The Connection between Galaxies and Dark Matter in the Young Universe

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Title:
The Connection between Galaxies and Dark Matter in the Young Universe
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1 online resource (147 p.)
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english
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Martinez Manso, Jesus
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Astronomy
Committee Chair:
GONZALEZ,ANTHONY HERNAN
Committee Co-Chair:
SARAJEDINI,VICKI LYNN
Committee Members:
HAMANN,FREDERICK,III
GUZMAN,RAFAEL
FRY,JAMES N

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Subjects / Keywords:
cosmology -- galaxies
Astronomy -- Dissertations, Academic -- UF
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Astronomy thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
The main goal of this work is to better understand how dark matter influences the formation and evolution of galaxies, from an observational perspective. I explored questions such as: how strongly do galaxies trace the dark matter field? At large scales, is dark matter density the only factor in galaxy formation? At small scales, how does the halo mass affect the efficiency of star formation within? I focused this study on the matter structures at high redshift, were measurements in the literature are still scarce and our knowledge largely incomplete. To shed light on the galaxy-halo connection, I present an analysis of the angular clustering of high-redshift galaxies in the recently completed 94 square degree Spitzer-SPT Deep Field survey. Applying flux and color cuts to the mid-infrared photometry efficiently selects galaxies at z~1.5 in the stellar mass range 10^{10}-10^{11}M_sun, making this sample the largest used so far to study such a distant population. Halo occupation distributions were fit to the data, finding a prominent peak in the stellar-to-halo mass ratio at a halo mass of log(M_halo)=12.44\pm0.08, 4.5 times higher than the z=0 value. This supports the idea of an evolving mass threshold above which star formation is quenched. In order to test how galaxies trace the matter distribution at large scales, I computed the cross-correlation between the z~1.5 galaxies and the cosmic microwave background convergence map from the South Pole Telescope. The best fit yielded a galaxy bias b^{g\kappa}=2.0\pm0.2, which is fully consistent with the value from the galaxy auto-correlation, b^{gg}=2.2\pm0.1. Therefore, the spatial distribution of galaxies is mostly driven by the large-scale matter density. In addition, I performed a test of the stellar masses of 4 galaxies at z=1 in the EGS field. These galaxies were previously found to be so small and massive that it posed a problem in terms of their evolution to match low redshift relations. I took GTC optical spectra of these galaxies, finding that they have dynamical masses ~6 smaller than previously thought. This alleviates the evolutionary problem that this sample represented.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jesus Martinez Manso.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: GONZALEZ,ANTHONY HERNAN.
Local:
Co-adviser: SARAJEDINI,VICKI LYNN.

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THECONNECTIONBETWEENGALAXIESANDDARKMATTERINTHEYOUNG UNIVERSE By JESUSMARTINEZ-MANSO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014

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c 2014JesusMartinez-Manso 2

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Dedicado,decoraz on,amiprimaCristina 3

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ACKNOWLEDGMENTS IwouldliketothankAnthonyGonzalezforgivingmetheopportunitytoworkon thisprojectandhelpmedevelopitthroughouttheyears.Hehasbeenanexcellent adviserandIdeeplyappreciatethetrustandlibertythathehasalwaysgrantedme. Atapersonallevel,Ihighlyvaluehisconstantwilltolisten,understandandsupportall requestsleadingtoimprovemyresearchexperienceandbuildmyprofessionalfuture. IwouldalsoliketoexpressmywarmestgratitudetoRafaelGuzman,mymaster's projectadviser.Igreatlyenjoyedourlengthydiscussionsaboutscience,whichopened mymindtounderstandsomeofthefundamentalquestionsinastronomy.Iamtruly gratefulforthetimehesharedwithmeandhiscontagiousoptimisimanddetermination. Ithasbeenapleasuretolearnfromhimasresearcherandaperson,forwhichIalso ndgreatadmiration. ManythanksaswelltoVickiSarajedini,FredHamann,RafaelGuzmanandJames Fryforputtingthetimeandefforttobepartofmythesiscommittee.Finally,Iwouldlike toacknowledgethehelpandsupportIhavereceivedfromcollaboratorssuchasGil Hoder,MatthewAshby,AdamStanfordandDuncanHanson. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 2MATCHINGGALAXIESANDHALOSAT z =1.5 ................. 16 2.1Background ................................... 16 2.2Datasets ..................................... 21 2.3ControlSample ................................. 23 2.3.1MainRedshiftDistribution ....................... 24 2.3.2DenitionofSubsamples ........................ 26 2.3.3StellarMasses ............................. 28 2.4Two-pointClustering .............................. 35 2.5PlacingGalaxiesInHaloes .......................... 38 2.5.1TheHaloOccupationDistribution ................... 38 2.5.2RedshiftScaling ............................ 42 2.6HODModelFits ................................ 45 2.7TheStellar-to-haloMassRatio ........................ 49 2.7.1ComparisontootherResultsat z =1.5 ............... 50 2.7.2EvolutionwithRedshift ......................... 53 2.8SatelliteGalaxies ................................ 57 2.8.1SatelliteFraction ............................ 57 2.8.2The M 1 / M min Relation ......................... 57 2.8.3PhysicalMechanismsforaMass-dependentEvolution ....... 60 2.9GalaxyBias ................................... 61 2.9.1ComparisontoWakeetal. ....................... 64 2.9.2ComparisontoOtherStudies ..................... 65 2.9.3BiasofCentralGalaxies ........................ 67 2.10Summary .................................... 68 3CROSSCORRELATIONOF z =1.5 GALAXIESWITHTHECOSMICINFRARED BACKGROUND ................................... 71 3.1Background ................................... 71 3.2DatasetsandMaps ............................... 72 3.3Measuredpowerspectra ........................... 73 3.4Theory ...................................... 75 5

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3.4.1Two-pointSpectrum .......................... 75 3.4.2Halomodel ............................... 76 3.4.3Haloluminosity ............................. 79 3.5Fittingapproach ................................ 80 3.6Resultsanddiscussion ............................ 81 3.7Summary .................................... 83 4CROSSCORRELATIONOF z =1.5 GALAXIESWITHCMBLENSING .... 88 4.1Background ................................... 88 4.2Datasets ..................................... 89 4.3Maps ...................................... 90 4.4Theory ...................................... 91 4.5Cross-correlation ................................ 92 4.6Discussion ................................... 94 5DYNAMICALMASSOF z =1 GALAXIES ..................... 97 5.1Background ................................... 97 5.2Dataset ..................................... 98 5.2.1SampleSelection ............................ 98 5.2.2ObservationsandReduction ...................... 98 5.2.3VelocityDispersions .......................... 100 5.2.4StellarPopulations ........................... 101 5.3DiscussionandSummary ........................... 101 6CONCLUSIONS ................................... 108 APPENDIX APHOTOMETRICSIMULATIONS .......................... 110 A.1SimulationProcedures ............................. 110 A.2PhotometricBias ................................ 115 A.3Completeness ................................. 116 BCOSMOSVSEGSASREFERENCESAMPLES ................. 118 CINTEGRALCONSTRAINT ............................. 121 DTHEHALOMODEL ................................. 124 ELOWREDSHIFTBUMP ............................... 129 FNOPRIORONNUMBERDENSITY ........................ 133 REFERENCES ....................................... 135 BIOGRAPHICALSKETCH ................................ 147 6

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LISTOFTABLES Table page 2-1SamplepropertiesandHODts .......................... 36 2-2Measuredangularcorrelation ............................ 39 2-3Best-tSHMRparameters .............................. 53 3-1Best-tCIBhalomodelparameters ......................... 83 5-1Physicalparametersoftheobserved z =1 compactgalaxies .......... 102 7

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LISTOFFIGURES Figure page 2-1PhotometricscatterofSSDFsources ....................... 24 2-2DistributionofIRACcolorvs.redshift ........................ 29 2-3DensitymapoftheSSDFeldforgalaxiesinourlargestsample ........ 30 2-4Effectofthecolorcutintheredshiftdistribution .................. 31 2-5Redshiftdistributionsofoursamples ........................ 32 2-6Redshiftevolutionofthemedianstellarmass ................... 33 2-7Stellarmasshistogramsofoursamples ...................... 34 2-8RedshiftdistributionoftheLimberkernel ...................... 46 2-9Measuredangularcorrelationfunctions ...................... 48 2-10Stellar-to-halomassratio .............................. 54 2-11ComparisonoftheSHMRslopeswithotherauthors ............... 55 2-12Redshiftevolutionofthe M 1 / M min ratio ....................... 56 2-13Bias, M 1 / M min ratioandsatellitefractionfromHODts .............. 62 2-14 M 1 / M min ratiovs.cumulativegalaxynumberdensity ............... 63 2-15ComparisonofHODbiastovaluesfromtheliterature .............. 66 3-1GalaxyandCIBmaps ................................ 74 3-2Angularpowerspectrabetween z 1.5 galaxiesandCIBmaps ........ 75 3-3MarginalizedposteriorlikelihoodsfortheparametersinourCIBhalomodel. 83 3-4Redshiftevolutionof M p ( z ) intheCIB ....................... 84 3-5RedshiftdistributionofCIBemission ........................ 85 3-6Redshiftevolutionofthecosmicstarformationratedensity ........... 86 4-1Galaxyandlensingconvergecemaps ....................... 91 4-2Galaxyandlensingredshiftkernels ......................... 93 4-3Galaxy-lensingcross-powerspectrum ....................... 95 5-1Spectraofthe z =1 compactgalaxies ....................... 103 8

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5-2Stellarmass-sizerelationat z 1 ......................... 104 A-1Photometricbias ................................... 111 A-2Photometricdetectioncompleteness ........................ 112 B-1ComparisonofHODresultsobtainedwithCOSMOSandEGScontrolsamples 120 C-1Angularcorrelationfunctionfrommockmaps ................... 123 E-1SuperCosmosR-bandhistogram .......................... 131 E-2Redshiftdistributionwithoutlow-redshiftbump .................. 132 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THECONNECTIONBETWEENGALAXIESANDDARKMATTERINTHEYOUNG UNIVERSE By JesusMartinez-Manso May2014 Chair:AnthonyH.Gonzalez Major:Astronomy Themaingoalofthisworkistobetterunderstandhowdarkmatterinuences theformationandevolutionofgalaxies,fromanobservationalperspective.Toshed lightonthegalaxy-haloconnection,Ipresentananalysisoftheangularclustering ofhigh-redshiftgalaxiesintherecentlycompleted94deg 2 Spitzer -SPTDeepField survey.Applyinguxandcolorcutstothemid-infraredphotometryefcientlyselects galaxiesat z 1.5 inthestellarmassrange 10 10 # 10 11 M ,yieldingthelargestsample usedsofartostudysuchadistantpopulation.Halooccupationdistributionsweret tothedata,ndingaprominentpeakinthestellar-to-halomassratioatahalomass of log( M halo / M )=12.44 0.08 ,4.5timeshigherthanthe z =0 value.Inaddition, Icross-correlatedthisgalaxysamplewithfar-infrared Herschel maps,inorderto directlylinkstarformationactivitywithdarkmatterhalos.Ifoundthatthestarformation efciencyofthesehalosincreasessteeplytowardshigherredshifts.Thecombination oftheseresultssupportstheideaofanevolvingmassthresholdabovewhichstar formationisquenched. Inordertotesthowgalaxiestracethematterdistributionatlargescales,Icomputed thecross-correlationbetweenthe z 1.5 galaxiesandthecosmicmicrowave backgroundconvergencemapfromtheSouthPoleTelescope.Thebesttyieldeda galaxybias b g =1.3 0.3 ,whichisnotconsistentwiththevaluefromthegalaxy auto-correlation, b gg =2.2 0.1 .Thisisasurprisingandunexpectedresult,and 10

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Ihavenotbeenabletodeterminewhetherithasaphysicaloriginoritisduetoan unaccountedsystematiceffect. Inaddition,Iperformedatestofthestellarmassesof4galaxiesat z =1 inthe EGSeld.Thesegalaxieswerepreviouslyfoundtobesosmallandmassivethatit posedaproblemintermsoftheirevolutiontomatchlowredshiftrelations.ItookGTC opticalspectraofthesegalaxies,ndingthattheyhavedynamicalmasses 6smaller thanpreviouslythought.Thisalleviatestheevolutionaryproblemthatthissample represented. 11

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CHAPTER1 INTRODUCTION Thecurrentparadigmofstructureformationestablishesthat,asacontinuous processintime,thecolddarkmattereldcollapsesintogravitationallyboundhaloes ( White&Frenk 1991 ; Springeletal. 2006 ),whichbecomeprogressivelyvirializedand growbytheaccretionofsurroundingmatter.Thisleadstohierarchicalbuild-up,where themostmassivestructuresarethelasttobeassembled.Atthesametime,baryonic gasboundtoahalocanloseitsangularmomentumbyradiativecoolingandfalltowards thecenterofthepotentialwell,whereitcancondenseandformstars( Silk 1977 ; Rees &Ostriker 1977 ; White&Rees 1978 ; Fall&Efstathiou 1980 ; Blumenthaletal. 1984 ). Thus,galaxiesmustforminsidedarkmatterhaloes,andastrongconnectionbetween thetwoisexpectedtopersistthroughouttheirevolution.Thissuggeststhatthehalo massmightregulatethebaryonicprocesses,andthereforethegalaxy-darkmatterlink mightholdthekeytomanyoftheobservedpropertiesofgalaxies. Thereisfairlygoodunderstandingofthedarkmatterdynamicsdowntodwarf-sized haloes,whichhasallowedforaccurate N -bodysimulationstobeperformed( Springel etal. 2005 ; Schayeetal. 2010 ; Klypinetal. 2011 ).Thesehavebeenusedby semi-analyticmethods( White&Frenk 1991 ; Springeletal. 2001 ; Crotonetal. 2006 ; Boweretal. 2006 )thattakehalomergertreesandapply adhoc analyticrecipes ofgalaxyformation.However,theyfailintheirpredictionsofanumberofobservables, especiallyathigh-redshift( Guoetal. 2011 ; Trujillo-Gomezetal. 2011 ).Otherstudies usingfullsimulationsincludingbaryonicprocessesalsoexist( Cen&Ostriker 1992 ; Katzetal. 1996 ; Springel&Hernquist 2003 ; Kere setal. 2009 ; Vogelsbergeretal. 2013 ),buttheseworksstillfaceimportantchallengesrelatedtothecomplexityofgas physics. Therefore,precisemeasurementsofthegalaxy-haloconnectionathighredshift aremuchneededinordertounderstandtheprocessesthatdrivegalaxyformation 12

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andevolution.Calculatinghalomassesonanindividualbasisisingeneralavery complicatedtask,butthereareobservationalapproachesbasedonlargenumber statisticsthatareabletoprovideanaveragequantity.Oneofthemisspatialgalaxy clustering,inconnectionwiththeassumptionthatgalaxiesmustresideinhaloes. Thestatisticalpropertiesofhaloesaregivenbysimulations,andtheobservedgalaxy clusteringcanbemodeledbythewaythatgalaxiespopulatethem.InChapter 2 ,we performaclusteringstudyofalargesampleof z 1.5 galaxies.Wemodelhowthese galaxiesaredistributedinsidehaloes,andestablishtherelationbetweenstellarandhalo massacrossalargedynamicrange.Themeasurementofthisrelationhasimportant consequencesregardingtheefciencyofstarformationasafunctionofhalomass, whichwendtobedifferentthanwhathasbeenreportedatlowredshift. Amoredirectinsighttotheconnectionbetweenstarformationandhalomass canbeobtainedbyclusteringmeasurementsoffar-infraredmaps.Theemissionat thesewavelengthscomesfrominterstellardustingalaxiesthathasbeenheatedby starformation,andthereforedirectlytracesthestarformationratedensity.Thepower ofclusteringmeasurementswiththistypeofdataresidesinthelargedynamicrange ofgalaxymassitprobes(anisotropiesaresensitivetocontributionsfromunresolved galaxies)anditsreachinredshift( z 4 ).InChapter 3 ,wecomputetheangular cross-correlationbetweenourgalaxydataand250,350and500 mmaps.Then,we applyahalomodeltotheclusteringsignal.Thisapproachplacesimportantconstraints ontheevolutioninthecosmicstarformationdensityandthestarformationefciencyin halos. Thisparadigmofstructureformationpredictsthatgalaxiestracethedarkmatter distributionatlargescales.Thismeansthat,onaverage,thegalaxyoverdensity indifferentregionsoftheUniverseshouldscaleinproportiontothedarkmatter overdensity.InChapter 4 ,wecheckthisassumptionbyperforminganindependent testbasedontheclusteringcrosscorrelationofthegalaxiesfromChapter 2 andamap 13

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oflensingconvergenceoftheCMB.ThelatterisderivedfromdatafromSouthPole Telescopebyotherauthors,andrepresentstheamountofintegratedmatteralongthe lineofsightupto z =1000 .Thus,theoverdensegalaxystructuresshould,inprinciple, overlapwithregionsofhightotalmassdensity.Wetestifthestrengthofthiscorrelation isconsistentwiththatpredictedfromtheinterpretationofgalaxy-galaxyclusteringbeing uniquelydrivenbyhalostatistics. Anotheraspectofthehighredshiftgalaxypopulationthatneedstobemore thoroughlytestedisthecalculationofstellarmasses.Theyareusuallyderivedapplying stellarpopulationmodelstotheobservedbroadbandspectralenergydistributions (SED)ofgalaxies( Bruzual&Charlot 2003 ; Maraston 2005 ),incombinationtoother assumptionssuchastheinitialmassfunction( Chabrier 2003 ; Kroupa 2001 ).Thereare stilllargeuncertaintiesinthesecalculations( Conroyetal. 2009 ; Mancone&Gonzalez 2012 ),speciallyforyoungpopulations,andthusformostgalaxiesat z > 1 .InChapter 5 ,weinvestigatedtherobustnessofstellarmassvalueswithintheparticularcaseofthe evolutionintherelationbetweenstellarmassandgalaxyradius.Severalobservational studiesintherecentyearshavereportedastrongevolutioninredshiftofthestructural parametersofgalaxies.Atxedmass,galaxiesweremuchsmallerinthepast,typically afactorof2at z =1 withrespecttolocalmeasurements.Numericalsimulationshave beenabletoreproducereasonablywellthisgrowthwithdryminormergersforthe averagepopulation.However,thereisanon-negligiblesubsetofhighz galaxieswith suchsmallradiithatnoplausiblemechanismcanevolvethemtohavesizescomparable towhatisseeninthelocalUniverse.Thus,wetooktheapproachoftestingwhether thistensioncouldbepartiallyexplainedbyasystematicerrorinthecalculationofstellar masses.Weselectedfourofthisextremegalaxiesat z =1 fromthemorphological catalogof Trujilloetal. ( 2007 ),whichincludesstellarmassesfromSEDtting.By measuringtheirspectrawiththeGTCtelescopeandderivingvelocitydispersions,we 14

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wereabletocalculatevirialmassesandfoundthattheseweremuchsmallerthanthe stellarmasses,suggestingthatthelatterhadbeenpreviouslyoverestimated. Chapter 2 andalltheappendiceshavebeensubmittedforpublicationandChapter 5 ispublished. 15

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CHAPTER2 MATCHINGGALAXIESANDHALOSAT Z =1.5 2.1Background Manyobservationalstudieshavemeasureddarkmatterhalomassesinordertond correlationswiththepropertiesofthegalaxiestheyhost.Variousworkshaveutilized gravitationallensingofbackgroundobjects( Mandelbaumetal. 2006 ; Gavazzietal. 2007 ; Boltonetal. 2008 ; Augeretal. 2010 ; Cacciatoetal. 2009 2013 ; Velander etal. 2011 ),virialtemperaturesderivedfromX-rays( Linetal. 2003 ; Lin&Mohr 2004 ; Peterson&Fabian 2006 ; Hansenetal. 2009 )anddynamicsofsatellites( Moreetal. 2009 2011 ).Thesemethodshaveachievedhighaccuracy,butarealsoobservationally expensivetocarryoutonlargesamplesandforsmallhaloes,whichlimitsthestatistical strengthandrangeofapplication.Alessdirectbutmorecomprehensivemethodof linkinggalaxiestohaloesisabundancematching( Conroyetal. 2006 ; Vale&Ostriker 2006 ; Mosteretal. 2010 ; Guoetal. 2010 ; Behroozietal. 2010 ; Mosteretal. 2010 2013 ),whichusesthemergertreesfrom N -bodydarkmattersimulationsasinputand assumesthatthehalomassisthemaindeterminantofgalaxyluminosityandstellar mass.Thebasicideaistocumulativelymatchobservedgalaxyluminosityfunctionsand halomassfunctionsbyplacingprogressivelylessluminousgalaxiesinlessmassive haloes.Bydesign,thismethodreproducestheluminosity(orstellarmass)function,and isabletopredicttheclusteringofgalaxiesinmanycases( Conroyetal. 2006 ; Conroy& Wechsler 2009 ; Mosteretal. 2010 ). Directmeasurementsofgalaxyclusteringareanotherpowerfulwaytoconnect galaxieswiththeunderlyingdarkmatterdistribution.Asafunctionofphysicalseparation r ,clusteringiscommonlymeasuredintheformofthetwo-pointspatialcorrelation function ( r ) (SCF;Peebles1980).Therelationbetweenthedistributionsofgalaxies anddarkmattercanbeparametrizedthroughthegalaxybias b g ( Kaiser 1984 ; Coles 1993 ; Fry&Gaztanaga 1993 ; Mo&White 1996 ; Kauffmannetal. 1997 ; Sheth& 16

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Tormen 1999 ; Tinkeretal. 2005 2010 ),whichisgivenbythescalingbetweenthe SCFsofthesetwoelds: g ( r z )= m ( r z ) b 2 g ( r z ). (21) TheSCFofdarkmatterdependsonthecosmology,andcanbeprescribedanalytically giventhoseparameters( Eisenstein&Hu 1999 ; Smithetal. 2003 ).Thus,thebiasof agalaxysampleisdirectlydeterminedbyitsSCF.Ingeneral,thebiasdependsonthe spatialscaleandredshift( Fry 1996 ; Moscardinietal. 1998 ; Tinkeretal. 2005 ; Moster etal. 2010 ),sincegalaxiesanddarkmatterdonotevolveintheexactsamemannerin timeorspace.Themeasurementof b g ( r z ) canthereforerevealaprecisedescriptionof theconnectionbetweengalaxiesanddarkmatter. Manystudiesuptointermediateredshifts( z < 1 )haveinvestigatedgalaxy clusteringwithsamplesselectedindifferentways( Phlepsetal. 2006 ; Zhengetal. 2007 ; Coiletal. 2008 ; Blakeetal. 2008 ; Brownetal. 2008 ; McCrackenetal. 2008 ; Meneuxetal. 2008 2009 ; Simonetal. 2009 ; Rossetal. 2010 ; Foucaudetal. 2010 ; Abbasetal. 2010 ; Zehavietal. 2011 ; Matsuokaetal. 2011 ; Wakeetal. 2011 ; Jullo etal. 2012 ; Leauthaudetal. 2012 ; Hartleyetal. 2013 ; Mosteketal. 2013 ; dela Torreetal. 2013 ; Donosoetal. 2013 ).Themostcommonconclusionisthatclustering strengthiscorrelatedwithluminosity,redcolorandmorphology(towardsearly-type). Galaxiesontheextremeofthesepropertiesarehighlybiasedandthereforetheylivein massivehaloes. Theseconclusionscanbeobtainedjustbyanalyzingtheoverallamplitudeofthe bias.However,thepreciseformofthisobservableasafunctionofspatialseparation containsmoreinformationabouttheinnerstructureofthehaloes.Thehalooccupation distribution(HOD)isasimpleparametricframeworktoaccuratelymodelthebias( Ma &Fry 2000 ; Seljak 2000 ; Peacock&Smith 2000 ; Scoccimarroetal. 2001 ; Cooray &Sheth 2002 ; Berlind&Weinberg 2002 ; Berlindetal. 2003 ; Kravtsovetal. 2004 ; 17

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Zhengetal. 2005 ).Itconsidersgalaxiestobeeithercentralsorsatellites,andthe numberofthesethatahalocanhostisfullydeterminedbythehalomass. OneoftheadvantagesoftheHODframeworkisthatitsparametershaveaclear physicalmeaning,andthuswhenttingtheclusteringonecangainadeeperinsight intotheconnectionbetweenthegalaxiesandtheirhosthaloes.Forexample,theHOD frameworkcandirectlyrelatetheaveragestellarmassofthecentralgalaxiestoa particularhalomass.Asshowninmanystudiesat z =0 # 1 ,theratioofthesemasses ishighestaroundahalomassof 10 12 M ( Zhengetal. 2007 ; Yangetal. 2012 ; Zehavietal. 2012 ; Leauthaudetal. 2012 ; Reddicketal. 2013 ; Behroozietal. 2013 ; Mosteretal. 2013 ; Wangetal. 2013 ).Thisimpliesthatthereisacharacteristichalo masswheregalaxyformationhasbeenmoreefcient.Thequalitativeexplanationfor thisisthatatlowhalomassesthegravitationalpotentialisnotdeepenoughtohaltthe expulsionofgasduetostellarwinds( Bensonetal. 2003 ),whilehigh-masshaloeshave heateduptheintra-halomediumbygravitationalheatingandAGNfeedback( Croton etal. 2006 ; Boweretal. 2006 ; vandeVoortetal. 2011b )sothatinfallinggasgets heavilyshockedandcannoteasilycoolandcondense( Birnboim&Dekel 2003 ; Dekel &Birnboim 2006 ; Kere setal. 2005 2009 ).Thesetwotrendscanbereducedtoa comparisonbetweendynamicalandgascoolingtimesinhaloes,suchthat dyn >>" cool forlowmassesand dyn <<" cool forhigh-masses.Apossibleconsequenceisthat thepeakhalomass M peak isrelatedtoacharacteristicquenchingmass M q thatsets dyn " cool ( Neisteinetal. 2006 )andmarksatransitionbetweenstarformingand quenchedhaloes.Indeed,massiveredgalaxieswithlittlestarformationhavebeen showntoliveinmassivehaloes( Coiletal. 2008 ; Zehavietal. 2011 ),supportingthe ideaoftheredsequenceofgalaxiesarisingwhentheybecomequenched( Crotonetal. 2006 ; Boweretal. 2006 ).Thisblue/reddichotomyispresentinthenearbyUniverse ( Kauffmannetal. 2003 2004 ; Baldryetal. 2004 ),andstartsitsbuild-uparound z 2 ( Belletal. 2004 ; Cooperetal. 2006 ; Muzzinetal. 2013a ; Wangetal. 2013 ).Thus, 18

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whenhaloesbecomelargeenough,theyquenchtheirstarformation.Aconsequenceof thisisthatthemostmassivegalaxiestodayhavenosignicantongoingstarformation. Thiseffecthasbeencalledarcheologicaldownsizing( Cowieetal. 1996 ; Juneauetal. 2005 ; Conroy&Wechsler 2009 ),andisalsoinferredfromthelackofevolutioninthe massiveendofthestellarmassfunction( P erez-Gonz alezetal. 2008 ; Marchesinietal. 2009 ; Muzzinetal. 2013a ).HODmodelshaveshownthatthestellar-to-halomassratio (SHMR)evolvesinthesensethatthepeakmovestolowerhalomasseswithincreasing time,atleastsince z 1 ( Couponetal. 2012 ; Leauthaudetal. 2012 ).Thistrend hasbeenpredictedtopersistupto z =2 byextensionsofHODthatuseconditional stellarmassfunctions( Yangetal. 2003 2012 ; Wangetal. 2013 )andabundance matchingstudies( Mosteretal. 2013 ; Behroozietal. 2013 ).Apossiblemechanism forthiswouldinvolveevolutionin M q ,whichissupportedbytheideathattheuniversal gasfractiondropswithtimeandthereforestarformationbecomesmoredifcultwith timeatxedhalomass( vandeVoortetal. 2011b a ).However,thisisstillamatterof debate( Conroy&Ostriker 2008 ; Tinker&Wetzel 2010 ).Forinstance, Leauthaudetal. ( 2012 )presentevidenceinfavorofthisevolutionbeingsetbyquenchingbelowacritical galaxy-halomassratioinsteadofacriticalhalomass.Suchamechanismwouldalso shifttheSHMRtowardlowermasseswithtime. Wehavedescribedthebasicprocessesthatcandetermine M peak ,basedon thecomparisonof dyn and cool asafunctionofhalomass.Thisbasicmodelcanbe extendedtoincludemodesofgalacticoutows,whicharethendirectlyconstrained bytheobservedslopeoftheSHMR.Thestellarmassgrowthofagalaxyisheavily regulatedbytheexpulsionofgas,whichcouldbemainlysourcedbysupernovae feedback( Murrayetal. 2005 ).Thestellarmasslossrate, M ,canbebrokendown intwocontributions:pressure-supportedenergyinjection(energy-drivenwinds)and coherentmomentumtransfer(momentum-drivenwinds).Theenergyandmomentum depositionrates, E and P ,canberelatedfromrstprinciplestothemassviaaproxyof 19

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thekineticvelocityeld, # W : M $ E / # 2 W and M $ P / # W .Thissuggeststhatgalaxies withlowvelocityelds,andthereforelowmasses,mayhavetheiroutowsdominatedby energy-drivenwinds( Dutton&vandenBosch 2009 ).Thus,alargercontributionfrom thistypeofwindwouldresultinasteeperlowmassslopeoftheSHMR. Athighmasses(andhigh # W ),theseargumentswouldpointtoadominance ofmomentum-drivenwinds.However,thewindsinthisregimearealsosourcedby radiativeAGNfeedback,whichisexpectedtohaveastrongcontribution( Vogelsberger etal. 2013 ).Inaddition,alargemergerratebetweencentralgalaxieswillresultina atteningoftheSHMR( Leauthaudetal. 2012 ).Withalltheseprocessesatplay,the high-massslopeislessstraightforwardtointerpretthanthelow-massone,butitcanstill offerimportantconstraintsonthiscombinationofmechanisms. GalaxyclusteringcombinedwithHODmodelingprovideparticularlysolidmeasurements oftheSHMRwhenevertheselectionofgalaxiesspanstherelevantrangeofstellar masses.At z 1.5 ,suchmeasurementshaveproventobeverydifcultgiventhe lackoflargevolume-limitedsamples. Wakeetal. ( 2011 )usethe0.25deg 2 NEWFIRM survey( vanDokkumetal. 2009 ),butthelownumberstatisticsmadeitdifculttomap theturnoveroftheSHMR.Inthisstudy,weusea94deg 2 mid-infraredsurveytoselect galaxieswithstellarmassesrangingfrom 10 10 # 10 11 M andtanHODmodeltothe angularcorrelationfunction.Wepresentthemostrobustmeasurementtodateofthe peakoftheSHMRat z =1.5 Inaddition,theHODyieldsparticularlystrongconstraintsonthesatellitepopulation ofagivengalaxysample.Wedeterminewhatfractionofthegalaxiesaresatellites, andhowtheabundanceofthesedependsonthehalomass.Moreover,wemeasurea proxyfortheoccurrenceofgalaxypairsofsimilarmass,andndthatitmildlydecreases towardhighluminosities.Althoughwedonotachievearobustdetection,thisrepresents theoppositetrendtowhatisseenatlow-redshift.Theprocessesthatproducethis 20

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relationshiparestronglytiedtotheaccretionandmergereventsbetweengalaxiesand haloes,aswellasthequenchingofstarformationinsatellites. Thechapterisorganizedasfollows.InSection 2.2 ,wedescribealldatasets thatareused.InSection 2.3 wedescribehowweadaptredshiftandstellarmass distributionsfromareferenceoptical+mid-IRsurvey.InSection 2.4 wedenethe two-pointclusteringstatisticandthemethodusedtocomputeit.InSection 2.5 wedescribethemodelthatlinksgalaxiestohaloes.InSection 2.6 ,weexplainthe ttingprocedureoftheHODtotheobservedclustering.InSections 2.7 2.8 and 2.9 wediscusstheresultsobtainedregardingtheSHMR,thesatellitegalaxiesandthe large-scalebias,respectively.WeendwithashortsummaryinSection 2.10 .Forthe readerthatisonlyinterestedintheresults,werecommendreadingSections 2.7 and beyond. Additionally,weincludeseveralappendiceswheremanyofthedetailsarecovered. Appendix A presentsacalibrationofsystematiceffectsinthephotometry.Appendix B comparestheresultsobtainedfromusingdifferentreferencecatalogstodrawredshift andstellarmassdistributions.Appendix C calculatesthesystematicoffsetinthe clusteringamplitudeduetothegeometryofthesurvey.Appendix D presentsthe formalismofthehalomodel.Appendix E investigatestheremovaloflow-redshift sourcesfromthesampleusingopticaldata.Appendix F exploresdifferentchoicesof freeparametersusedintheHODtstotheclustering. Throughoutthischapterweusethefollowingcosmology: m =0.27 ! =0.73 and H 0 =70kms 1 Mpc 1 .AllmagnitudesareintheVegasystemandmassesareinunitsof M 2.2Datasets Ourmaindatasetisthe Spitzer SouthPoleTelescopeDeep-FieldSurvey(SSDF; Ashbyetal.2013b),a93.8deg 2 photometricsurveyusingtheIRAC3.6and4.5 m bands(hereafter[3.6]and[4.5]).Themosaicshaveanominalintegrationtimeof120 21

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seconds.WeusedSourceExtractor(SExtractor;Bertin&Arnouts1996)indualimage mode,detectinggalaxiesin[4.5]andextractingtheuxfromxed4 ## aperturesinboth IRACchannels.Theseapertureuxeswerethencorrectedtototaluxesusinggrowth curvesfromisolatedpointsourcesfoundinthemosaics.Adetaileddescriptionofthe surveyandapublicphotometriccatalogarepresentedin Ashbyetal. ( 2013 ).However, hereweuseadeeperprivatecatalogandaccountforfaint-endphotometricbiasand detectioncompleteness(seeAppendixA).Wedeterminethe5 # limitin[4.5]tobe18.19 mag,inagreementwith Ashbyetal. ( 2013 ). Weusethenear-infrared2MASSPointSourceCatalog( Skrutskieetal. 2006 ) toidentifyandremovesourcesbrighterthan K s ( AB )=12 mag,mostofwhichare likelytobestars.Inaddition,wevisuallyinspectedsomeofthesesourcesintheIRAC mosaicsanddeterminedanempiricalrelationbetweentheir K s -bandmagnitudeandthe maximumradiuswheretheir4.5 muxcausedaclearsuppresioninthedetectionof nearbysources.Thisrelationwasthenappliedtotherestofthe K s -selectedsampleand theresultingradiiwereusedtomaskallSSDFsourcesenclosedwithinfromthemain catalog.Forreference,theradiicorrespondingto K s ( AB )=8 and K s ( AB )=12 sources were41 ## and8.4 ## respectively.Wealsomaskedoutlowcoveragegapsinthesurvey, yieldinganaleffectiveareaof88.8deg 2 Finally,inordertobetterunderstandtheredshiftdistributionofourIRAC-selected sampleintheSSDF,weusepubliccatalogsintwootherregionsofthesky:the COSMOS-UltraVistaeld(hereafterCOSMOS,Muzzinetal.2013)andtheExtended GrothStrip(hereafterEGS,Barroetal.2011a,b).Thesetwosurveyshavepublically accessibleIRACphotometry,photometricredshifts,andstellarmasses.Inthefollowing Sectionwedescribehowweusedthesecatalogstoinfertheredshiftandmass distributionsofSSDFsamples. 22

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2.3ControlSample Thisstudyrequiresknowledgeoftheredshiftandstellarmassdistributionofthe SSDFgalaxysample.However,ourmaindatasetistoolimitedtoobtainreliablevalues fortheseobservables.Therefore,thestrategyistoimportthisinformationfroma referencesurveythatcontainsopticaldataandIRACphotometrywithahigheraccuracy. WeconsiderthecatalogsfromCOSMOSandEGS,whichincludephotometricredshifts andstellarmasses.WewilladoptCOSMOSastheducialdatasetbecauseitislarger andhasbetterstatistics,andinAppendix B weshowhowourresultsdonotchange signicantlywhenusingEGSinstead.Thereferencecatalogisdegradedtobecome acontrolsamplewhosephotometricerrorsmatchthoseoftheSSDF.Then,applying thesameIRACselectioninbothSSDFandthecontrolsampleallowsustomatch thederiveddistributionsofredshiftandmass.AbriefdescriptionoftheCOSMOS photometrycanbefoundinAppendix B Foreverysourceinthereferencecatalog,wehave [4.5] magnitudes, [3.6] # [4.5] colors,photometricredshiftsandstellarmasses.ThegoalistoinfertheSSDF distributionsoftheseparametersbydegradingthereferencephotometry,whichisdone usingtheSSDFphotometricerrors.WecalculatethescatterinSSDFmagnitudesand colorsasafunctionofthesesamevariables,usingtheresultsfromthephotometric simulationsdescribedinAppendix A .ThesescatterprolesareshowninFigure 2-1 .At xed[4.5]magnitude,thescatterincolorincreasesforlargercolorssincetheseimply fainter[3.6]magnitudes.Inthecaseofthereferencesample,sinceitis2magnitudes deeperthanSSDF(seeAppendix B ),wecansafelyconsideritsphotometricscatteras negligibleincomparison. Thedegradationofthereferencecatalogintoacontrolsampleconsistsof transformingthespecicvalues(e.g.,magnitude)ofeachsourceinthecatalog intoGaussianprobabilitydensityfunctions(PDFs).ThesePDFsaredenedinthe parameterspaceofapparentmagnitude[4.5]( M ), [3.6] # [4.5] color( C )andphotometric 23

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15 16 17 18 19 [4.5] mag 0 05 0 10 0 15 0 20 0 25 0 30 0 35 [4 5] 0 2 0 4 0 6 0 8 1 0 [3.6] [4.5] mag 0 05 0 10 0 15 0 20 0 25 [3 6] [4 5] Figure2-1. PhotometricscatterofSSDFsourcesderivedfromthesimulationsin Appendix A Left: Standarddeviationin[4.5]magnitudes.Thereishigher scatterforfaintersources. Right: Standarddeviationin[3.6]-[4.5]color(4 ## diameteraperture).Dashed,dotted,solidanddash-dottedrepresentxed [4.5]inputmagnitudesof15,16,17and18,respectively.Largercolorsimply fainter[3.6]magnitudes,whichisreectedasamildincreaseinthescatter. redshift( z phot ): P ( M C z phot ) .Thecentroidsaregivenby $ i = M i C i z i phot ,which correspondtotheparametervectorsofthesourcesinthereferencecatalog.The standarddeviationsare $# i = # # i M # i C # i z phot $ .Thersttwocomponentsin $# arethe functions # M ( M ) and # C ( M C ) ,whichareshownbythecurvesinFigure 2-1 .The redshiftcomponentdoesnothaveacounterpartintheSSDFcatalog,butweapplya variableredshiftsmoothingkernelequivalentto100comovingMpc,inordertolterthe effectoflarge-scalestructure.Thisamountsto # z phot =0.02 # 0.1 withinourredshift range. 2.3.1MainRedshiftDistribution Ifweconsidergalaxieswithapparentmagnitudeswithinsomebracket M ,wecan computethedistributionincolorandredshiftspace K ( C z ) ofthecontrolsample: 24

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K ( C z )= 1 N ref N ref % j =1 & M dm P ( m C z ; $ j $# j ). (22) Here,wehavemarginalizedeachindividualPDFover M andsummedtheminthe resultingspaceof ( C z phot ) ,usingthereferencecatalog(subscript"ref").Asimilar proceduretoderivefullredshiftdistributionsbasedontheBayesiancombinationof individualredshiftlikelihoodfunctionswasperformedby Brodwinetal. ( 2006b a ).The normalizationof K ( C z ) isthetotalnumberofsourcesinthereferencecatalog, N ref Figure 2-2 showstheapplicationofEquation(1)for M % 15 < [4.5] < 18.6 ,which arethelimitsforourfullSSDFsample(seeSection 2.3.2 ).Thetoppanelcorrespond tothecolorversusredshiftdistributionfromtherawreferencecatalog.Thelowerpanel showsthecontrolsample,whichishowSSDFsourcesareexpectedtobedistributed. Forcomparison,wehavealsoplottedagalaxyevolutionarytrackforasinglestellar populationwithsolarmetallicityandformationredshiftof z f =3.5 ,computedusingthe Bruzual&Charlot ( 2003 )modelswith Chabrier ( 2003 )IMF. Thereisaclearcorrelationbetweencolorandredshiftat z > 0.6 .Thisoccurs becausegoingfrom z =0.6 to z =2 ,theIRACbandsmapthegalaxyspectrumacross thestellarbumpatrest-frame H -band.Thisresultsinamonotonicchangeinobserved colorwithin z =0.6 # 2 .Thestarformationhistorymakesonlyaweakcontribution.An insightfuldescriptionofthisphenomenologycanbefoundin Muzzinetal. ( 2013b ).We cantakeadvantageofthiseffecttoselectgalaxiesinredshiftusingacolorcut.Alower colorthresholdneedstobehighenoughtoreject z < 0.3 galaxies(seeFigure 2-2 ), whilealsokeepinganumberofhigherredshiftsourcesthatislargeenoughtomeasure arobustclusteringsignal.Anupperthresholdisalsonecessary,sinceveryredcolors [3.6] # [4.5] 1 arecharacteristicofactivegalacticnuclei( Sternetal. 2005 ).Thebest compromiseisacolorcutof 0.6 < [3.6] # [4.5] < 0.8 ,asshowninFigure 2-2 Withagiven C ,wecanderivetheredshiftdistributionofsources: 25

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% ( z )= 1 N ref N ref % j =1 & M & C dmdc P ( m c z ; $ j $# j ). (23) Notethat % ( z ; M C ) dz isequalto1onlywhen M and C representthefull rangesspannedbythereferencesources.Wedenotesuchdistributionas % full ( z ) ,while theonecorrespondingtothecolorselection C % 0.6 < [3.6] # [4.5] < 0.8 isdenotedas % cut ( z ) 2.3.2DenitionofSubsamples OurmainsciencesampleofSSDFgalaxiesisdeterminedbytheapparent magnitudeandcolorcutsof 15 < [4.5] < 18.6 and 0.6 < [3.6] # [4.5] < 0.8 .The rstofthesecutsimposesanuppermagnitudelimitatthe80%completenesslevel(see Appendix A.3 ),andthesecondistunedtoselectgalaxiesathigh-redshiftwhileavoiding AGN.AdensitymapofthisselectioncanbeseeninFigure 2-3 ,representingasliceof theUniverseat z 1.5 Wefurthersplitthemainsampleinto13subsamples,withfaintlimitsovertherange [4.5]=16.2 # 18.6 instepsof0.2mag.Thebrightlimitis [4.5]=15 inallofthem. Wedothisinsteadofaselectionwithindifferentialmagnitudebinsbecausethehalo occupationframeworkpresentedinSection 2.5 requirescumulativesamplesinorderto linkhalomassesandgalaxymasses.Wenotethatthisapproachcarriesthedrawback ofproducingacorrelationbetweenthedifferentsamples.Thiscorrelationisstrong betweenneighboringsamples,butnotdominantotherwise.Duetothesteepvariation ofthestellarmassfunction,anygivensampleismostlycomprisedbygalaxiesclose toitslow-massthreshold,makingtheirclusteringlesssensitivetothemostmassive population(seeMatsuokaetal.2011). Thephotometricscatterincreasesforfaintersamples.Thus,wecalculatethe redshiftdistribution(seeEquation 23 )foreachsample,obtainingsetsof % k full % k cut where k =1 # 13 isthesampleindex(goingfrombrightesttofaintest).Wecanalso denethenumberdensitycompletenessas f k N ( z )= % k cut ( z ) / % k full ( z ) ,whichdetermines 26

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thefractionofgalaxiesasafunctionofredshiftthatthecolorcutretains.Figure 2-4 showsacomparisonof % cut % full for k =13 (thelargestsample).Atthepeakofthe color-cutdistributionwehavethat f k N 0.3,andwewillusethisfactortoscaleupand correctthenumberdensity(seebelow).Figure 2-5 shows % k cut forthesmallestand largestsamples(i.e.,brighterandfainterthresholds, k =1,13),whereeachcurveis shownnormalizedto1.Wealsoderivecosmicvarianceerrorsusingtheprescriptions from Mosteretal. ( 2011 ),whicharebasedonanalyticalpredictionsofdarkmatter structuregivenaparticularsurveygeometry(seealsoBrodwinetal.2006a).The peakintheseredshiftdistributionsisconsistentlyaround z =1.5 inallsamples.In general,thesamplesconsistofa z & 1 populationthathasapproximatelythesame absoluteluminosityandstellarmass(seeSection 2.3.3 ),plusa z 0.3 contributionof "contaminant"galaxiesthatareintrinsicallymuchlessluminous.Thesecontaminants represent12%(37%)ofallgalaxiesinourfull(brightest)sample.Whensettinga brighteruxthreshold,thehigh-redshiftpopulationbecomeslessdominantsince thesegalaxiesareclosertotheturnoveroftheluminosityfunction.Theconsequence ofthisisthecleartrendwherebrightersampleshaveastrongerlow-redshiftbump. ThecontributionofthelattertotheclusteringismodeledinthefollowingSections. Alternatively,weshowinAppendix E thatourresultsremainunchangedifinsteadwe employshallowopticaldatatoremovemostofthelow-redshiftsources. Withtheseredshiftdistributions,wecancalculatethespatialnumberdensityof observedgalaxiesatthepivotredshift z p 1.5 .Here,weusetheSSDFsurveyarea andtheeffectivenumberofobservedsourcesinthesubsamples, N k obs .Thisnumberis derivedbysummingtheinverseofthecompletenessvalueforallgalaxies,usingthe relationfromFigure A.1 .Then,thetruenumberofgalaxieswithin z p & z / 2 canbe writtenas N k true = N k obs f k N ( z p ) % k cut ( z p ) % k cut ( z # ) dz # & z (24) 27

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Thesampledvolumereadsas V = dV ( z p ) dz & z = c 2 ( z p ) H ( z p ) & z (25) where ( z ) isthecomovingradialdistance, H ( z ) istheHubblefunction, c isthespeed oflightand !=0.0271 steradiansisthesolidanglesubtendedbythesurvey.Hence, thenumberdensityat z p resultsin n k g = N k true V (26) NotethatthisquantityistheresultofcombiningtheSSDFobservednumbercounts(via N k obs )andthecolorfractionsofthecontrolsample.Table 2-1 showsthevaluesofthese numberdensitiesforallsamples. 2.3.3StellarMasses Thestellarmassesinthereferencecatalogarealsoretrievedtoconstructour controlsample.WeusethosebasedonBruzual&Charlot(2003,hereafterBC03) stellargrids, Chabrier ( 2003 )IMFand Calzettietal. ( 2000 )dustextinction.Unless otherwisenoted,allstellarmassesaregivenundertheseprescriptions. Forthepurposesofthischapter,weneedtocalculatestellarmassesfortwo differentselectionsofgalaxies.Oneisthemedianmassofallgalaxieswithineach sample,derivedateveryredshiftbin, M full .Thismasswillbeusedtoderivearedshift scalingofthegalaxybiasinSection 2.5.2 .Theother, M lim ,isthemedianmassofthe galaxiesatthepivotredshift( z p =1.5 )andatthemagnitudelimitofeachsample.This isthestellarmassthatwillbelinkedinSection 2.7 toaparticularhalomass. Tocalculate M full ,rstweassignweightstothegalaxiesinthereferencecatalog, representingthefractionalcontributionofeachonetoaselectiondenedwithinsome apparentmagnitudeandcolorintervalsandforaspecicredshift: ( j ( z )= & M & C dmdc P ( m c z ; $ j $# j ), (27) 28

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Figure2-2. [3.6]-[4.5]colorvs.redshiftforgalaxieswith 15 < [4.5] < 18.6 ,basedonthe COSMOSreferencecatalog.Thehorizontalwhitelinesindicatethecolor selectionthatweapplytoourSSDFsamples.Thepurplecurveisthe evolutionarytrackofagalaxyformedat z f =3.5 usingtheBC03modelwith aChabrierIMF,shownforcomparison. Top: Distributionsoftheraw referencecatalog. Bottom: ReferencecatalogdegradedtomatchtheSSDF photometricproperties,derivedfromEquation 22 29

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Figure2-3. DensitymapoftheSSDFeldforgalaxiesinourselectedsample: 0.6 < [3.6] # [4.5] < 0.8 and 15 < [4.5] < 18.6 .Thiscorrespondstoaredshift selectionaround z 1.5 .Unitsaregalaxiespersquarearcminute.Masking hasbeenappliedtobrightstarsandlowcoveragegaps,yieldinganalsize of88.8squaredegrees. where 0 < ( j < 1 and j isanindexthatgoesthroughallgalaxiesinthereference catalog.Next,wecancombinetheseweightswiththeindividualstellarmasses M j to derivethemedianstellarmass.Aweightedmedianrepresentsthemassvaluewhere theweightedintegralofthemassdistributionaboveandbelowitisthesame.Thus,we canwritethemedianstellarmassofthesampleas M full =Median ( $ M ;weights= $( ) (28) wherewehaveomittedtheimplicitdependenceon z .Theweighteddistributionof massesfollowscloselyalog-normaldistribution.Thus,Equation 28 returnsalmost thesamevalueastheweightedmeanof log $ M .Hence,wecanadoptthestandard deviationfromthelatterdistribution: 30

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0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 Redshift 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 ( z ) [4.5] < 18.6 [4.5] < 18.6, 0.6 < [3.6] [4.5] < 0.8 Figure2-4. RedshiftdistributionoftheCOSMOS-basedcontrolsampleusingthe faintestselection(15 < [4.5] < 18.6, k =13 ).Thedashedlinesrepresentthe additionalcolorcutselection.Thecolorcutimposesaselectionaround z 1.5 ,althoughitonlykeepsaboutonethirdofthetotalnumbercountsat thatredshift. # M = log M j # log M full " 2 ( j ( j (29) whichistypically 0.2 dex.Eventhoughthisscatterisratherlarge,theselog-mass distributionsaresingle-peakedandapproximatelysymmetric,sotheirmeanvalueis well-denedandphysicallymeaningful.Weusethescattertoestimatetheerrorinthe meanas # M = # M ( N ind (210) with N ind = ( ( j ) 2 ( j 2 (211) 31

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0 0 0 5 1 0 1 5 2 0 2 5 3 0 Redshift 0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 ( z ) / dz ! ( z ) 0.6 < [3.6] [4.5] < 0.8 [4.5] < 16.2 [4.5] < 18.6 Figure2-5. NormalizedredshiftdistributionsoftheCOSMOS-basedcontrolsample usingthefaintest(15 < [4.5] < 18.6,orangetriangles)andbrightest (15 < [4.5] < 16.2,bluecircles)uxthresholdswiththecolorcut.Brighter samplethresholdsinduceahighercontributionoflow-redshiftsources(see text). Here, N ind representstheeffectivenumberofindependentelementsintheensemble. Thisnumberisproportionaltothesumofcontributingweightsandinverselyproportional totheirscatter.Itequalsthetotalnumberofelementsinthereferencecataloginthelimit of ( j % 1 Figure 2-6 shows M full ( z ) forallsamples.TheerrorsarefromEquation( 210 )and thesolidcurveisa5thorderpolynomialttothepointsofthefaintestsample.Weuse thatcurveplusanoffsettotthedatafromtherestofthesamples,sinceitbecomes noisieratbrighterlimits.Here,wetakeadvantageofthefactthatstellarmassscales withuxapproximatelyinalinearmanner.Itisclearfromthegurethatthemassis tightlycorrelatedwiththeredshiftofobservationwithin z =0 # 1.5 .Beyondthat,the 32

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0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 Redshift 9 0 9 5 10 0 10 5 11 0 11 5 12 0 log M full [4.5] < 16.2 [4.5] < 18.6 Figure2-6. Redshiftevolutionofthemedianstellarmassinourbrightestandfaintest samples.Thelowersolidcurveisapolynomialttothepointsfromthe latter.Thosecorrespondingtothebrightsamplearenoisier,soweoffsetthe lowersolidcurvetomatchthem.Thisisphysicallymotivatedbythe approximationthatmassscaleslinearlywithux. relationattensoutsignicantly.Thereasonforthisisthatat z & 1.5 ,the[4.5]band samplestherisingspectralslopeofthestellarbump( Muzzinetal. 2013b ).Thisoffsets thek-correctioninawaythatgalaxiesofacertainintrinsicnear-infraredluminosityhave asimilarapparent[4.5]magnitudeacrossarangeofredshift.Aconsequenceofthis isthatany[4.5]limitedsamplebecomesroughlystellarmasslimitedat z > 1.5 (see alsoFigure14inBarroetal.2011b).Nonetheless,wedonotattempttotakeadvantage ofthiseffectbyaveragingstellarmassesathigh-redshift.Themodelinginthisworkis basedonwell-denedmedianmassesasafunctionofredshift,independentoftheform ofthatredshiftdependence.However,theatteningofthiscurvedoesbenetourstudy tosomeextent.SincethereisaninherentuncertaintyinhowwellrepresentedtheSSDF 33

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9 0 9 5 10 0 10 5 11 0 11 5 log M lim 0 00 0 05 0 10 0 15 0 20 0 25 0 30 dN d log M [4.5] = 16.2 [4.5] = 18.6 Figure2-7. Normalizeddistributionsofthe z =1.5 stellarmassatthebrightestand faintestmagnitudelimitsofoursamples. dataiswiththecontrolsample,itisconvenientthatthestellarmassesarenaturally moreconstrainedthanacasewheretheyhadastrongredshiftdependence. Wecancalculate M lim at z p inananalogousway,consideringaselectionwithin C Thecorrespondingweightsare ) j ( M )= & C dc P ( M c ; z p $ j $# j ), (212) andreplacing ( j with ) j inEquations(7-10)givesusthemedianmass M lim ,along withitserror.Anexampleofthestellarmasshistogramsthatareobtainedusingthese weightscanbeseeninFigure 2-7 .Theyareshownforthebrightestandfaintest magnitudelimitsofoursamples.Thesedistributionsaresymmetricandhavea well-denedmean.Thescatterisgenerallylarge,withvaluesaround 0.2dex. However,theerrorinthelogarithmicmean(seeEquation( 210 ))istypicallyquite 34

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small 0.03dex.ThevaluesoftheselimitingmassesaredisplayedinTable 2-1 .Thus, wehavecalculatedthemedianmassofallgalaxiesinoursamplesasafunctionof redshift,andthemedianmassofgalaxiesaroundthepivotredshiftateachsample magnitudelimit. 2.4Two-pointClustering Givenapopulationofgalaxiesinathree-dimensionalspace,onecandenethe jointprobabilityofndingtwosuchobjectsinvolumeelements & V 1 & V 2 separatedbya distance r ( Peebles 1980 ; Phillippsetal. 1978 ): & P ( r )= N 2 (1+ g ( r )) & V 1 & V 2 (213) Here, N isthedensityofgalaxiesand g istheSCF,whichquantiestheclustering strengthoftheeldasafunctionof r .TheSCFcanalsobeinterpretedasthe differentialprobabilityofndingtwoobjectsseparatedbyagivendistance,withrespect tothecaseofarandomdistribution. TheSCFforagalaxypopulationcanbedirectlycomputediftheindividualdistances (redshifts)tothosegalaxiesareknown.However,inourcasewearelimitedtoindividual skypositionsandtheensembleredshiftdistribution.Therefore,weareinterestedin theangularcorrelationfunction(ACF),whichistheprojectionoftheSCFontothe2D sphere.AnalogouslytotheSCF,theACFrepresentsthedifferentialprobabilitywith respecttoarandomdistributionofndingtwogalaxiesseparatedbyaparticularangle. TheACFisrelatedtotheSCFthroughtheLimberprojection( Limber 1953 ),which integratestheSCFalongthelineofsightusingthenormalizedredshiftdistribution % ( z ) asaweightkernel( Phillippsetal. 1978 ; Couponetal. 2012 ): ( + )= 2 c & $ 0 dzH ( z ) % 2 ( z ) & $ 0 dy g ( r = + y 2 + D 2 c ( z ) + 2 ), (214) where D c ( z ) istheradialcomovingdistance, H ( z ) istheHubblefunction, c isthespeed oflightand + istheangularseparationgiveninradians. 35

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Table2-1. SamplepropertiesandHODts. Column1: Upperlimitingmagnitudeforeachsample.Thelowerlimitisxed at15mag. Column2: TotalnumberofobservedSSDFsources,correctedforcompleteness. Column3: Estimatedfractionbetweenobservedandtruenumberofsourcesat z p Column4: Numberdensityat z p correctedby f N .Unitsare 10 4 Mpc 3 .Errorsarederivedbythemethodof Mosteretal. ( 2011 ). Columns5-6: Medianstellarmassesof z = z p galaxiesatandabovetheuxlimitofthesample,respectively. Columns7-11: ParametersfromtheHODts. M # 1 arebest-tvalues,therestarederivedparameters.Thereducedchi-squared isgivenby 2 # = 2 / (28 # 1) .Since =1 ,onecandirectlycompute M 1 = M # 1 + M 0 ,with log M 0 =0.76log M # 1 +2.3 [4.5]limit N obs f N n g log M lim log M full log M # 1 log M min b g f sat 2 # 16.2177130.430.4 0.110.93 0.0211.05 0.0214.28 0.0913.17 0.053.95 0.130.06 0.010.33 16.4293850.420.7 0.210.86 0.0210.99 0.0214.03 0.1113.00 0.063.57 0.120.09 0.010.74 16.6472420.401.3 0.310.80 0.0110.91 0.0213.80 0.0912.84 0.053.28 0.080.12 0.011.27 16.8725060.402.2 0.410.72 0.0110.83 0.0113.62 0.0912.70 0.053.04 0.070.15 0.021.79 17.01058010.403.2 0.610.63 0.0110.74 0.0113.51 0.0812.57 0.042.85 0.050.15 0.011.93 17.21467730.404.5 0.810.54 0.0110.65 0.0113.39 0.0912.47 0.052.72 0.060.16 0.021.76 17.41953460.396.0 1.010.46 0.0110.57 0.0113.28 0.0812.38 0.052.62 0.040.18 0.011.22 17.62494440.387.6 1.310.36 0.0110.48 0.0113.20 0.0812.30 0.042.52 0.040.18 0.011.04 17.83080640.379.3 1.510.26 0.0110.37 0.0113.12 0.1012.24 0.062.46 0.040.20 0.020.90 18.03707350.3511.1 1.810.16 0.0110.27 0.0113.06 0.0912.17 0.052.40 0.040.20 0.021.03 18.24356720.3413.0 2.110.06 0.0110.18 0.0113.00 0.0712.12 0.052.35 0.030.21 0.010.95 18.45032120.3215.0 2.39.97 0.0110.08 0.0112.95 0.0712.07 0.052.30 0.030.22 0.010.88 18.65751310.3017.2 2.69.87 0.019.99 0.0112.89 0.0712.03 0.052.27 0.030.23 0.010.56 36

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Inordertomeasure ( + ) weusetheestimatorpresentedin Hamilton ( 1993 ),which countsthenumberofgalaxypairswithrespecttothoseofarandomsampledistributed inthesamegeometry: ( + )= RR( + )GG( + ) GR 2 ( + ) # 1, (215) whereGG,GRandRRaretotalnumberofgalaxy-galaxy,galaxy-randomand random-randompairsseparatedbyanangle + .Wehavealsotestedtheestimator from Landy&Szalay ( 1993 ),whichreturnsresultsthatarepracticallyindistinguishable fromthoseusingEquation 215 InordertoaccountforthecompletenessshowninFigure A.1 ,wemakeasmall generalizationofEquation 215 .Insteadofcountingallpairswithvaluesof1,weuse aweightedschemewhereeachpairofsources , iscountedasaproductofweights $ % .Randomsourceshave =1 ,andgalaxieshaveweightsequivalenttotheinverse ofthecompletenessvalueatitsapparentmagnitude.Wecountspairsbybruteforcein discreteangularbinsusingthegraphicsprocessingunit(GPU)onadesktopcomputer. Wehavedevelopedourowncode,whichyieldscomputationtimesoftheorderof 1000timesfasterthanusingaCPU-basedrunwith16cores.Ourcodeiswrittenin PyCUDA 1 ,whichisaPythonwrapperoftheCUDA,theprogramminglanguagethat interfaceswiththedevice. TheestimatorinEquation 215 implicitlyassumesthattheaveragegalaxydensity ofthesurveyisthesameastheall-skyvalue.However,sincethesurveyisasmall fractionofthesky,itsdensityishigher(structuresclustermoretowardsmallerscales) andthisresultsinasystematicsuppressionof ( + ) .Wecorrectforthiseffect,even 1 documen.tician.de/pycuda/ 37

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thoughitisnotsignicantforourresults.DetailscanbefoundinAppendix C .The valuesofthecorrectedACFforallsamplesaredisplayedinTable 2-2 ErrorsintheACFareestimatedwiththejackknifetechnique,whichusesthe observeddataandisveryeffectiveinrecoveringthecovarianceof ( + ) between differentscales.First,theentiresampleisdividedinto N jack =64 spatialregionsofequal size.Then,thecorrelationisrun N jack times,eachoneexcludingoneofthoseregions fromthesample.Thevalueoftheestimatoristheaverage ( + ) ofthoseiterationsand thecovariancebetweenangularbinsisgivenby( Scrantonetal. 2002 ) C jk = N # 1 N N % i =0 [ i ( + j ) # ( + j ) ][ i ( + k ) # ( + k ) ] (216) Wealsocomparethejackknifeerrorswiththoseobtainedfrommocksimulations,which aredescribedinAppendixA.Wendthatbothsetsoferrorshaveagoodagreement, withdifferencesaround20%.Althoughourmocksimulationsonlycoverlarge-scales, thesystematicdifferencesbetweenmockandjackknifeerrorsarenotexpectedtovary signicantlyacrossdifferentscalesforaprojectedstatisticlike ( + ) ( Norbergetal. 2009 ). 2.5PlacingGalaxiesInHaloes Thegalaxybias b g (seeEquation 21 )encodesalltheinformationthatcanbe extractedfromthetwo-pointgalaxydistribution,givenaparticularcosmology.Thus,our aimistoconstructaprecisemodelof b g andadjusttheresultingcorrelationfunction tomatchtheobservedclusteringofgalaxies.Themainideabehindthismodelisto assumeahalodistributionandplacegalaxiesinhaloesaccordingtoasetofsimple rules,asexplainedbelow. 2.5.1TheHaloOccupationDistribution ThedistributionofdarkmatterhaloesundertheCDMparadigmhasbeen well-studiedbothphenomenologicallyandthroughsimulations( Ma&Fry 2000 ; Cooray &Sheth 2002 ; Berlind&Weinberg 2002 ),leadingtoahalomodelwherethehalo 38

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Table2-2. Measuredangularcorrelationforalloursamples,whicharedenotedbytheirlimiting[4.5]magnitudeintherst row.Thesevalueshavebeencorrectedfortheintegralconstraint. + (degrees)16.216.416.616.817.017.217.4 0.00192.8 0.9 ) 10 0 1.9 0.4 ) 10 0 1.7 0.2 ) 10 0 1.5 0.2 ) 10 0 1.3 0.1 ) 10 0 1.2 0.1 ) 10 0 1.1 0.1 ) 10 0 0.00251.8 0.5 ) 10 0 1.6 0.3 ) 10 0 1.2 0.1 ) 10 0 1.1 0.1 ) 10 0 8.6 0.6 ) 10 1 7.5 0.4 ) 10 1 6.5 0.3 ) 10 1 0.00339.5 3.0 ) 10 1 9.3 1.6 ) 10 1 7.5 0.9 ) 10 1 6.6 0.7 ) 10 1 5.8 0.5 ) 10 1 5.3 0.3 ) 10 1 4.6 0.2 ) 10 1 0.00444.2 2.1 ) 10 1 5.1 1.2 ) 10 1 5.0 0.6 ) 10 1 4.3 0.3 ) 10 1 3.6 0.3 ) 10 1 3.3 0.2 ) 10 1 2.9 0.2 ) 10 1 0.00583.7 1.9 ) 10 1 3.6 0.8 ) 10 1 3.7 0.5 ) 10 1 3.6 0.3 ) 10 1 3.0 0.2 ) 10 1 2.5 0.2 ) 10 1 2.0 0.1 ) 10 1 0.00773.0 1.0 ) 10 1 1.5 0.5 ) 10 1 1.7 0.3 ) 10 1 1.6 0.2 ) 10 1 1.7 0.1 ) 10 1 1.4 0.1 ) 10 1 1.4 0.1 ) 10 1 0.01022.1 0.9 ) 10 1 1.9 0.4 ) 10 1 2.0 0.2 ) 10 1 1.7 0.1 ) 10 1 1.4 0.1 ) 10 1 1.3 0.1 ) 10 1 1.1 0.1 ) 10 1 0.01351.8 0.5 ) 10 1 1.6 0.3 ) 10 1 1.4 0.2 ) 10 1 1.3 0.1 ) 10 1 1.1 0.1 ) 10 1 1.0 0.1 ) 10 1 9.3 0.8 ) 10 2 0.01789.7 5.7 ) 10 2 1.0 0.2 ) 10 1 1.1 0.1 ) 10 1 1.1 0.1 ) 10 1 9.9 0.8 ) 10 2 8.8 0.7 ) 10 2 7.7 0.7 ) 10 2 0.02351.4 0.3 ) 10 1 1.3 0.1 ) 10 1 1.0 0.1 ) 10 1 9.3 0.9 ) 10 2 7.9 0.7 ) 10 2 7.4 0.5 ) 10 2 6.6 0.5 ) 10 2 0.03116.3 3.0 ) 10 2 8.5 1.9 ) 10 2 7.6 1.0 ) 10 2 7.5 0.7 ) 10 2 6.3 0.5 ) 10 2 5.7 0.4 ) 10 2 5.0 0.4 ) 10 2 0.04115.9 1.9 ) 10 2 6.8 1.1 ) 10 2 7.2 0.7 ) 10 2 6.8 0.7 ) 10 2 5.7 0.4 ) 10 2 4.9 0.3 ) 10 2 4.4 0.2 ) 10 2 0.05434.5 1.7 ) 10 2 5.3 1.0 ) 10 2 4.7 0.7 ) 10 2 4.3 0.5 ) 10 2 4.2 0.3 ) 10 2 3.8 0.3 ) 10 2 3.5 0.2 ) 10 2 0.07174.9 1.2 ) 10 2 3.9 0.8 ) 10 2 3.6 0.5 ) 10 2 3.5 0.3 ) 10 2 3.2 0.2 ) 10 2 2.8 0.2 ) 10 2 2.6 0.1 ) 10 2 0.09472.5 1.1 ) 10 2 2.6 0.6 ) 10 2 2.8 0.5 ) 10 2 2.7 0.3 ) 10 2 2.5 0.2 ) 10 2 2.3 0.2 ) 10 2 2.1 0.1 ) 10 2 0.12502.8 0.8 ) 10 2 2.7 0.5 ) 10 2 2.1 0.3 ) 10 2 2.2 0.2 ) 10 2 1.9 0.2 ) 10 2 1.7 0.1 ) 10 2 1.6 0.1 ) 10 2 0.16512.0 0.6 ) 10 2 1.8 0.4 ) 10 2 1.8 0.3 ) 10 2 1.7 0.3 ) 10 2 1.5 0.2 ) 10 2 1.4 0.1 ) 10 2 1.3 0.1 ) 10 2 0.21801.8 0.5 ) 10 2 1.3 0.3 ) 10 2 1.4 0.3 ) 10 2 1.4 0.2 ) 10 2 1.2 0.1 ) 10 2 1.0 0.1 ) 10 2 9.9 1.3 ) 10 3 0.28797.2 4.4 ) 10 3 1.1 0.3 ) 10 2 1.0 0.2 ) 10 2 9.8 2.0 ) 10 3 9.2 1.7 ) 10 3 8.1 1.3 ) 10 3 7.3 1.1 ) 10 3 0.38029.4 3.6 ) 10 3 8.3 2.2 ) 10 3 7.4 2.0 ) 10 3 6.6 1.7 ) 10 3 5.7 1.4 ) 10 3 5.2 1.0 ) 10 3 4.7 0.9 ) 10 3 0.50215.5 2.7 ) 10 3 4.4 2.1 ) 10 3 5.1 1.5 ) 10 3 4.6 1.2 ) 10 3 3.7 0.9 ) 10 3 3.3 0.8 ) 10 3 3.0 0.7 ) 10 3 0.66304.8 2.4 ) 10 3 3.2 1.8 ) 10 3 3.1 1.3 ) 10 3 3.3 1.2 ) 10 3 2.7 1.0 ) 10 3 2.3 0.8 ) 10 3 2.1 0.7 ) 10 3 0.87551.7 1.8 ) 10 3 2.6 1.4 ) 10 3 2.5 1.0 ) 10 3 2.3 0.9 ) 10 3 1.4 0.8 ) 10 3 1.5 0.6 ) 10 3 1.3 0.5 ) 10 3 1.15612.0 1.7 ) 10 3 1.7 1.2 ) 10 3 1.6 0.9 ) 10 3 1.0 0.8 ) 10 3 8.5 6.7 ) 10 4 7.7 5.8 ) 10 4 7.8 5.4 ) 10 4 1.5266-5.1 13.0 ) 10 4 1.3 9.4 ) 10 4 1.0 7.2 ) 10 4 -1.1 6.8 ) 10 4 -0.2 6.1 ) 10 4 -1.0 4.8 ) 10 4 1.0 4.5 ) 10 4 2.0158-7.4 9.5 ) 10 4 0.3 7.1 ) 10 4 -2.9 5.9 ) 10 4 -3.7 5.5 ) 10 4 -5.4 4.5 ) 10 4 -5.1 3.6 ) 10 4 -4.6 2.9 ) 10 4 2.6618-4.0 10.0 ) 10 4 1.4 7.1 ) 10 4 -2.7 6.0 ) 10 4 -4.2 4.9 ) 10 4 -2.6 4.3 ) 10 4 -1.9 3.9 ) 10 4 -3.0 3.3 ) 10 4 3.5148-9.4 9.3 ) 10 4 -1.2 0.5 ) 10 3 -1.0 0.4 ) 10 3 -7.4 4.3 ) 10 4 -4.7 3.3 ) 10 4 -4.4 2.9 ) 10 4 -4.2 2.5 ) 10 4 39

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Table 2-2 .Continued + (degrees)17.617.818.018.218.418.6 0.00199.0 0.4 ) 10 1 7.5 0.3 ) 10 1 6.5 0.3 ) 10 1 5.7 0.2 ) 10 1 5.0 0.2 ) 10 1 4.3 0.2 ) 10 1 0.00255.7 0.3 ) 10 1 5.0 0.2 ) 10 1 4.4 0.2 ) 10 1 3.9 0.2 ) 10 1 3.4 0.2 ) 10 1 2.9 0.1 ) 10 1 0.00334.0 0.2 ) 10 1 3.5 0.2 ) 10 1 3.0 0.2 ) 10 1 2.5 0.2 ) 10 1 2.3 0.1 ) 10 1 1.9 0.1 ) 10 1 0.00442.6 0.1 ) 10 1 2.3 0.1 ) 10 1 2.0 0.1 ) 10 1 1.8 0.1 ) 10 1 1.6 0.1 ) 10 1 1.4 0.1 ) 10 1 0.00581.9 0.1 ) 10 1 1.6 0.1 ) 10 1 1.4 0.1 ) 10 1 1.2 0.1 ) 10 1 1.1 0.1 ) 10 1 1.0 0.1 ) 10 1 0.00771.2 0.1 ) 10 1 1.1 0.1 ) 10 1 1.0 0.1 ) 10 1 9.0 1.0 ) 10 2 8.0 0.9 ) 10 2 7.1 0.9 ) 10 2 0.01021.0 0.1 ) 10 1 9.4 0.9 ) 10 2 8.6 0.8 ) 10 2 7.5 0.8 ) 10 2 6.7 0.8 ) 10 2 6.2 0.8 ) 10 2 0.01358.1 0.8 ) 10 2 7.4 0.8 ) 10 2 6.5 0.7 ) 10 2 5.8 0.7 ) 10 2 5.3 0.7 ) 10 2 4.8 0.7 ) 10 2 0.01786.5 0.6 ) 10 2 5.9 0.5 ) 10 2 5.3 0.5 ) 10 2 5.0 0.5 ) 10 2 4.4 0.5 ) 10 2 3.9 0.5 ) 10 2 0.02355.8 0.4 ) 10 2 5.3 0.4 ) 10 2 4.8 0.4 ) 10 2 4.1 0.3 ) 10 2 3.7 0.3 ) 10 2 3.3 0.3 ) 10 2 0.03114.5 0.3 ) 10 2 4.1 0.3 ) 10 2 3.8 0.3 ) 10 2 3.4 0.3 ) 10 2 3.0 0.3 ) 10 2 2.7 0.2 ) 10 2 0.04114.1 0.2 ) 10 2 3.4 0.2 ) 10 2 3.0 0.2 ) 10 2 2.7 0.2 ) 10 2 2.4 0.2 ) 10 2 2.1 0.2 ) 10 2 0.05433.1 0.2 ) 10 2 2.7 0.2 ) 10 2 2.4 0.1 ) 10 2 2.1 0.1 ) 10 2 1.9 0.1 ) 10 2 1.7 0.1 ) 10 2 0.07172.4 0.1 ) 10 2 2.0 0.1 ) 10 2 1.8 0.1 ) 10 2 1.6 0.1 ) 10 2 1.5 0.1 ) 10 2 1.3 0.1 ) 10 2 0.09471.8 0.1 ) 10 2 1.7 0.1 ) 10 2 1.5 0.1 ) 10 2 1.3 0.1 ) 10 2 1.2 0.1 ) 10 2 1.0 0.1 ) 10 2 0.12501.5 0.1 ) 10 2 1.3 0.1 ) 10 2 1.2 0.1 ) 10 2 1.0 0.0 ) 10 2 9.5 0.8 ) 10 3 8.4 0.7 ) 10 3 0.16511.1 0.1 ) 10 2 1.0 0.1 ) 10 2 9.6 1.1 ) 10 3 8.7 1.1 ) 10 3 7.7 1.0 ) 10 3 6.9 0.9 ) 10 3 0.21808.7 1.2 ) 10 3 7.8 1.0 ) 10 3 6.8 0.9 ) 10 3 6.2 0.7 ) 10 3 5.7 0.7 ) 10 3 5.0 0.6 ) 10 3 0.28796.3 1.0 ) 10 3 5.6 0.9 ) 10 3 5.0 0.8 ) 10 3 4.6 0.7 ) 10 3 4.1 0.7 ) 10 3 3.6 0.6 ) 10 3 0.38024.1 0.8 ) 10 3 3.7 0.7 ) 10 3 3.4 0.7 ) 10 3 3.0 0.6 ) 10 3 2.8 0.6 ) 10 3 2.5 0.5 ) 10 3 0.50212.7 0.6 ) 10 3 2.4 0.5 ) 10 3 2.3 0.5 ) 10 3 2.0 0.4 ) 10 3 1.8 0.4 ) 10 3 1.6 0.3 ) 10 3 0.66301.8 0.6 ) 10 3 1.5 0.5 ) 10 3 1.5 0.5 ) 10 3 1.2 0.4 ) 10 3 1.1 0.4 ) 10 3 1.0 0.3 ) 10 3 0.87551.0 0.5 ) 10 3 8.8 4.7 ) 10 4 8.3 4.3 ) 10 4 7.6 3.8 ) 10 4 7.2 3.4 ) 10 4 5.7 3.0 ) 10 4 1.15616.5 4.5 ) 10 4 6.6 4.2 ) 10 4 5.4 3.7 ) 10 4 5.6 3.3 ) 10 4 5.4 3.0 ) 10 4 4.5 2.6 ) 10 4 1.52661.2 3.7 ) 10 4 1.8 3.2 ) 10 4 1.7 2.7 ) 10 4 1.9 2.6 ) 10 4 1.7 2.3 ) 10 4 1.2 2.1 ) 10 4 2.0158-3.3 2.6 ) 10 4 -2.0 2.3 ) 10 4 -1.4 2.0 ) 10 4 -1.1 1.9 ) 10 4 -0.7 1.7 ) 10 4 -0.8 1.6 ) 10 4 2.6618-2.1 2.6 ) 10 4 -2.4 2.3 ) 10 4 -1.8 2.0 ) 10 4 -1.7 1.8 ) 10 4 -2.0 1.5 ) 10 4 -1.8 1.4 ) 10 4 3.5148-3.3 2.0 ) 10 4 -2.8 1.8 ) 10 4 -2.7 1.5 ) 10 4 -2.6 1.4 ) 10 4 -2.4 1.3 ) 10 4 -1.8 1.1 ) 10 4 40

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massfunction,thebias b h andthehalodensityprolearedeterminedbythehalo mass.TheHaloOccupationDistribution(HOD)isastatisticalframeworkthathasbeen developedtolinkthehalomodelwiththedistributionofgalaxies( Berlind&Weinberg 2002 ; Cooray&Sheth 2002 ; Kravtsovetal. 2004 ).TheHODismainlydescribedwith theprobability P ( N | M ) thatahaloofagivenvirialmass M hosts N galaxies;onecentral and N # 1 satellitesdistributedaccordingtoaNFWprole.Allgalaxiesarelinkedto somehalo,andtheoccupationisindependentoftheirformationhistoryandenvironment ( Zentneretal. 2005 ).Thisassumptionisgenerallyvalid,sincetheinducedchangesin thegalaxybiasduetoenvironmentareexpectedtobeonlyatthe 5%level( Croton etal. 2007 ; Zuetal. 2008 ),whiletheoveralluncertaintiesingalaxyclusteringstudies aretypicallylarger.Forourworkinparticular,themainsourceoferrorarisesfromthe uncertaintyintheshapeoftheredshiftdistribution,whichisexploredinAppendix B bycomparingresultsfromtheuseofCOSMOSandEGSasreferencecatalogs.The variationsingalaxybiasarearound10%andtheydonotalterqualitativelyanyofthe nalconclusions.Therefore,giventhattheenvironmentaleffectsinthegalaxybiasare expectedtobesmaller,weconsiderthemnegligibleforthecurrentpurposes. Theaveragedistributionofcentralgalaxiesasafunctionofhalomasscanwritten as( Zhengetal. 2005 2007 ): N c ( M )= 1 2 1+erf log M # log M min # log M ./ (217) Thisimpliesthat N c ( M min )=0.5 .Thus, M min setsastep-liketransitionwherehalfofthe haloesabovethismasswillhostacentralgalaxy,andthistransitionissmoothedbythe scatter # log M .ThenumberofsatellitesgalaxiesisdrawnfromaPoissondistributionwith mean N s ( M )= N c ( M ) M # M 0 M # 1 $ (218) 41

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andareassumedtofollowaNFW( Navarroetal. 1997 )densityprolefromthehalo center.Thefactor N c ( M ) accountsfortheconstraintthatonlyhaloeswithacentral galaxymayhostsatellites.Equation 218 representsapowerlaw,where setsthe steepness, M # 1 denesthetypicalmassscaleforthisdistributionbeingclosetounity and M 0 representsthemassbelowwhichthepower-lawiscutoff.Inaddition,onecan derivethecharacteristicmasswhereahalohostsexactlyonesatelliteonaverage, M 1 byimposing N s ( M 1 ) 1 andnotingthatgenerally N c ( M M # 1 )=1 .Inthecasewhere M 0 =0 itreducessimplyto M 1 = M # 1 ,andwhen =1 then M 1 = M # 1 + M 0 .The occupationdistributionofthetotalnumberofgalaxiesinahalocanbeexpressedasthe sumofthecentralandsatelliteterms: N ( M )= N c ( M )+ N s ( M ). (219) OtherHODderivedquantitiesaretheeffectivegalaxybias b e # g = 1 n g & dM dn ( M ) dM N ( M ) b h ( M ), (220) andthefractionofsatellitegalaxies f sat = 1 n g & dM dn ( M ) dM N s ( M ). (221) FurtherdetailsaboutthehalomodelusedherecanbefoundintheAppendix D 2.5.2RedshiftScaling Weaimtotthehalomodelatthepivotredshift z p =1.5 .However,thegalaxiesin oursampleshaveredshiftdistributionsthataretoobroadtobeneglectedoraveraged over(seeFig. 2-5 ).Thus,ourapproachistoproduce p g g ( z = z p ) ,andscaleitusing asimpleprescriptiontogenerate g atallotherredshifts.WethenuseEquation 214 to makearedshiftprojectionof g ( z ) onto ( + ) Thescalingweapplyisbasedonhowthelarge-scaleclustering(representedbythe two-haloterm 2 h g ,seeEquation D16 )varieswithredshift.Thischangeinamplitudeis 42

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drivenbythegrowthfactor G ( z ) ofthedarkmatterandthegalaxybias b g ( z ) .Hence,we canwrite g ( z )= G 2 ( z ) G 2 ( z p ) b 2 g ( z ) b e # g 2 p g (222) where b e # g and p g aresetat z p byconstruction.Here,wehavemadetheapproximation thattheentirecorrelationfunctioncanbescaledwithasinglefactor.However,the relativeamplitudeoftheoneandtwo-halotermsisknowntoevolve( Conroyetal. 2006 ; Watsonetal. 2011 ),inthesensethattypicallytheone-halotermismoreprominentat higherredshift.Wehavetestedhow ( + ) wouldchangeifweallowforsomedifferential redshiftscalingbetweentheoneandtwo-halotermsof g .Thiswasdoneapplyinga linearlyredshift-dependentfactortotheone-haloterm,inadditiontothegeneralscaling fromEquation 222 .Inthisway,belowandabove z p theone-halobecomesreduced andboosted,respectively.WendthatthishasaverylittleeffectontheresultingACF. Thisisbecausetheredshiftdistributionsofourgalaxiesaremorelesssymmetric,so thattherelativescalingoftheone-haloaboveandbelow z p isalmostcanceledwhen thesecontributionsaresummedtogether.Inreality,thisrelativescalingmighthavea morecomplexdependenceonredshift,butwebelievethatthelinearrepresentation weconsideredhereisadequategiventhesymmetricandpeakedformsofourredshift distributions.Thus,wendthatourmodelisnotsensitivetotheparticularevolutionof theone-halotermanddonotincorporateitinthedeterminationofourresults. Ourgeneralapproachistocalculatethebiasasafunctionoftheevolvingmedian stellarmass, b g ( z )= b g ( M full ( z )) .Forthispurpose,wemakeuseofthegalaxybias asafunctionofstellarmassandredshiftpresentedinMosteretal.(2010,hereafter M10), b M10 g ( M z ) .However,wedonotusetheirbiasvaluesdirectlysinceweneed toenforcethat b g ( z p )= b e # g ,i.e.,thebiasfunctionhastomatchtheHODbiasatthe redshiftofthet.Ourbiasfunctionisnormalizedtoholdthatconstraint,butthescaling atotherredshiftsisadoptedfromM10(foragivenstellarmass).Toaccomplishthis, rstwedenethestellarmass M # where b M10 g ( M # z p )= b e # g .Ideally, M # wouldbe 43

PAGE 44

equaltothemedianmassofthesamplefromSection 2.3.3 M full ( z p ) ,buttheydiffer. Thisisnotsurprising,sincethemodelinginM10isbasedonabundancematching, whichisdifferentfromourclusteringapproachandcanpotentiallyyielddifferingvalues ofthebias.Inaddition,somevariationsareexpectedgiventhedifferencesinmodels andcodesusedtoderivestellarmassesinM10andourreferencesample.However, theM10massesbythemselvesarenotrelevanttous,andtheysimplyrepresenta quantityorlabelthatlinksbrighterpopulationsofgalaxieswithalargerbias.Therefore, itissufcienttoassume apriori thatallthesemassesholdamonotonicrelationship withsampleluminosity,whichhasbeenprovencorrect aposteriori .Inotherwords, M # doesscalemonotonicallywith M full acrossallsamples.So,foragivensample,what wecalculateistheoffset log M =log M # # log M full at z p .Inthisway,weareableto "convert"ourstellarmassesintoM10masses.Ourbiasfunctionthenbecomes b g ( z )= b M10 g (log M full ( z )+" log M z ). (223) InSection 2.3.3 wecalculated M full ( z ) forallsamples.Forthebrightestones,the databecomesabitnoisyduetothelownumberdensity(seeFigure 2-6 ).This massdistributionisconsistentwithhavingaconstantshapeandvaryingitbysome normalizationthatscaleswiththemagnitudelimitofthesample.Thus,weadoptthe functionalformofthelargestsample M full ( z ; k =13) ,whichisgivenbythepolynomial tshowninFigure 2-6 .Thenormalizationofthisfunctiondoesnotneedtobetakeninto account,sinceitwillbeimplicitlyincorporatedin log M Thestellarmassdependenceofthebiashasanimportanteffecton ( + ) .Because thestellarmassofoursamplesislargerathigh-redshift(about10timeslargerat z = 1.5 thanat z =0.5 ;seeFigure 2-6 ),thebiaswillplaceastrongerweighttherethanat low-redshift.Thishelpstominimizethecontributionoftheundesiredlow-redshiftbump at z 0.3 .Additionally,thereareotherfunctionsweighingintheLimberprojection, whichisproportionalto H ( z )[ % ( z ) b g ( z ) G ( z )] 2 ,asinferredfromEquations 214 and 44

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222 .Figure 2-8 showsthecomparisonbetweentheredshiftdistribution % ( z ) andthe fullLimberkernel.Itisshownforthebrightestsamplebecauseitistheonewiththe highestfractionoflow-redshiftcontaminants.Intheend,thelow-redshiftcontribution totheclusteringisminimizedduetothedecreasein M full ( z ) ,whichsuppresses b g ( z ) Thiseffectisconvenientforouranalysis,sinceitmakestheclusteringpropertiesofour sampleshighlyrepresentativeofthe z 1.5 Universe.Moreover,inAppendix E we investigatehowthenalresultsareimpactedbytheuseofavailableopticaldatainthe SSDFeldtoremovelow-redshiftsources.Wendthatthechangesintheresultsare negligiblecomparedtokeepingthesesourcesandmodelingtheirweakcontributionto theclustering,asdoneinthisSection. ApossibleconcernatthispointisthattheresultsfromM10wouldbe"built-in"to oursthroughthecouplingwithEquation 223 .However,thenormalizationofthebias issetbyourowndata,anditistheredshiftmodulationthatweincorporatefromthese authors.Inaddition,wehaveexploredvariationsof b M10 g ( M z ) anddeterminedthat ourmodelisnotverysensitivetosuchchanges.AsseeninFigure 2-8 ,theredshift modulationplaysaroleinweighinggalaxiesat z =0.3 vs z =1.5 .Basically,any functionthatdown-weightsthelow-redshiftbumpwilldosoinamannerthatitbecomes quicklysubdominant.Wejustneedafunctionthatreectsapproximatelythevariationof thebiaswithstellarmassandredshift,whichispreciselywhatisprovidedbyscalings fromM10.Ourmodeldoesnotstronglydependonthedetailedformofthisfunction,and wehaveveriedthatourresultsarenotpre-setinasignicantwaybythoseinM10. 2.6HODModelFits Thettingprocedureisbasedonmaximizingthelikelihoodofthemodelgiventhe observable L ( mod | obs )= e & 2 ,with 2 = N % i =0 N % j =0 [ m ( + j ) # ( + j ) ] C 1 ij [ m ( + k ) # ( + k ) ] (224) 45

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0 0 0 5 1 0 1 5 2 0 2 5 Redshift 0 0 0 5 1 0 1 5 2 0 2 5 3 0 [4 5] < 16 2 Selection function ( z ) Lim b er k ernel H ( z )[ ( z ) b g ( z ) G ( z )] 2 Figure2-8. Comparisonofthenormalizedredshiftdistribution % ( z ) ofthebrighest sample(15 < [4.5] < 16.2),whichhasthemostprominentlow-redshiftbump, andthecorrespondingLimberkernel.They-scalingisarbitraryineither curve.IntheLimberprojection,thebiasfunctionbooststhecontributionof highz galaxies,sincetheyarealsomoremassive.Thiseffectminimizesthe contributionofthelow-redshiftbumptotheACF. Here, m and arethemodelandobservedACFs, C ij isthecovariancematrixfrom Equation 216 and N =28 isthenumberofangularbins.Thehalooccupationmodel weconsiderhasatotalof5parameters: M min M # 1 M 0 , and # log M .Eventhoughthe signal-to-noiseofourACFsisverygood( 11 # 31 # withrespecttothenullhypothesis), thefactthatitistheresultofprojectingtheSCFacrossawideredshiftdistribution reducesourconstrainingpowerontheHODmodel.Thus,toavoidover-fttingthedata, wechoosetoxanumberofparameters.WehaverunsetsofMonteCarloMarkov chainstoexplorethesensitivityofthemodeltodifferentchoicesofconstraints.To evaluatethissensitivity,weusetheAkaikecriterion( Akaike 1974 ),whichstatesthatan extrafreeparameterisjustiedonlywhenthenewbest-t 2 isreducedbyanamount largerthan2.Foreither # log M and M 0 ,thiscriterionisnotfulllled.Thus,wefollow 46

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Conroyetal. ( 2006 )andset log M 0 =0.76log M 1 +2.3 .Wealsox # log M =0.2 ,following anumberofstudiesthatsupporttypicalvalues > 0.15 ( Moreetal. 2009 ; Behroozi etal. 2010 ; Moreetal. 2011 ; Wakeetal. 2011 ; Mosteretal. 2013 ; Reddicketal. 2013 ; Behroozietal. 2013 ).Inthecaseof ,wehavethat 2 3 ,whichwouldmildly favorsettingitfree.However,thisparameterhasanintrinsicdegeneracywith M # 1 and whenleftfreetooat,thebesttvaluesshowasignicantstochasticcomponentin theirbehaviorwithrespecttosampleluminosity.Itcannotbeconstrainedaswellas M 1 andthuswedecidetoxittoacommonchoiceintheliteraturethatisalsosupported bysimulations, =1 ( Kravtsovetal. 2004 ; Zentneretal. 2005 ; Tinkeretal. 2005 ; Zhengetal. 2005 ; Zehavietal. 2011 ; Wakeetal. 2011 ; Leauthaudetal. 2012 ).None ofthenalconclusionsinthisworkchangewhetherornotweallow tovaryfreely. Additionally,Equation D ## 7 xes M min throughtheobservedgalaxynumberdensity, leaving M # 1 astheonlyparameterleftinthet.InAppendix F wecommentonhowthe resultingHODmodelchangesifweleavenearlyallparametersfreeinthet. Obtainingthebest-tvalueof M # 1 isstraightforward.Theerrorinthetcanbe estimatedfromthewidthofthelikelihooddistribution,butitdoesnotaccountfor departuresarisingfromcosmicvariance.Toaccountforthat,weperformasetof 100randomrealizationsoftheredshiftdistributionandnumberdensityat z p ,whichwe call % rd cut ( z ) and n rd g .WendthebestHODt M # 1 eachtime,alongwiththecorresponding derivedparameters.Eachredshift j -binin % rd cut, j isdrawnfromanormaldistributionwith mean % cut, j andstandarddeviationasinSection 2.3.1 (seeFigure 2-5 ).Thevaluefor n rd g isproducedinasimilarmanner;usinganormaldistributionwithmeanandstandard deviationequaltothevalueanderrorof n g inTable 2-1 .Thescatterinallparameters fromtherandomrealizationsisclearlydominantoverthatarisingfromthewidthofthe M # 1 likelihoodintheducialt,especiallyduetothevariationsin n g .Wecantherefore approximatethenalerrorsasthosefromtherandomrealizations. 47

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10 2 10 1 10 0 (degrees) 10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 w ( ) 16.2 16.8 17.4 18.0 18.6 Figure2-9. Observedangularcorrelationfunction(pointswitherrorbars)ofthe samples 15 < [4.5] < { 16.2,16.8,17.4,18.0,18.6 } .Thesolidcurves correspondtothebestmodelts.Anextradecadehasbeenaddedbetween consecutivecurvesforeasiervisualization.Theerrorbarsaredrawnfrom thediagonalelementsofthecovariancematrix(Equation 216 ).Ingeneral, neighboringpointsarepositivelycorrelated,whilethosefarapartare anti-correlated.ThisisaninherentpropertyoftheACFestimation( Norberg etal. 2001 ; Scrantonetal. 2002 ),butcanbeeffectivelytakenintoaccount viaametricoftheformofEquation 224 Figure 2-9 showstheobservedACFandthemodeltsforafewsamples.The valuesanderrorsforallrelevantparametersaregiveninTable 2-2 48

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2.7TheStellar-to-haloMassRatio Haloesofmassesequalto M min hostonaverage0.5centralgalaxieswith luminositiesgreaterthanthesamplethreshold(Equation 217 ). Zhengetal. ( 2007 ) showedanalyticallythatcentralgalaxieslivingintheseparticularhaloeshavea medianluminositycorrespondingtothelimitofthesample.Thislinkshalomasses withluminosities,albeitwithsomescatter 0.15 dex( Zehavietal. 2011 ; Couponetal. 2012 ).However,weareinterestedintheconnectionwithstellarmasses,whichalso haveawell-denedmeanandscatteratxedluminosity(seeSection 2.3.3 ).Hence, wecanlink M min to M lim ,withascatter( # log M )thatoughttobeclosetothequadratic sumofthescattersfromtheluminosityM min andluminosityM lim relations.Wemeasure thelattertobearound0.2dex,andtheformerisexpectedtobesimilar.Thus,thefact thatwex # log M =0.2 mightseemanunderestimation.However,aswewilldiscussin Appendix F ,anunconstrainedHODtdoesnotpreferlargervaluesforthisparameter. Also,thenalresultsdonotchangesignicantlybyincreasingittolargervaluesas0.4 dex.Wethusretainourchoiceandproceed. Thevaluesof M lim and M min foroursamplescanbefoundinTable 2-1 .Theirratio yieldstheSHMR,whichisplottedasafunctionofhalomassinFigure 2-10 forour differentsetsofstellarmasses.Theverticalerrorbarsareacombinationofthehalo massuncertaintyandtheerrorinthemedianstellarmasses(Equation 210 ),i.e.,it doesnotrepresentthescatterinstellarmassatxedhalomass.Itisinterestingthat theerrorbarsdonotgetnotablybiggerforbrightersamples,eventhoughtheACFsof thosearemuchnoisierandthestellarmasserrorsareindeedlarger.Thereasonisthat M min becomesprogressivelylesssensitivetotheHODtathigherluminosities.Thet isbasedon M # 1 ,whichfallsclosetothesteepdropofthehalomassfunction(Equation D1 )inthebrightsamplesandmakestheoverallHODmodelbeweaklyaffectedby thesatelliteoccupation(e.g.,Equation D7 ).Thus,theerrorcontributionfrom M # 1 is 49

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minimized,andthatfrom n g and M lim increases,keepingthetotalerrorroughlyconstant acrossthedifferentsamples. 2.7.1ComparisontootherResultsat z =1.5 ThereareseveralstudiesthathavetriedtoconstraintheSHMRat z > 1 ,basedon abundancematching( Mosteretal. 2013 ; Behroozietal. 2013 ),HODmodeling( Zheng etal. 2007 ; Wakeetal. 2011 ; Couponetal. 2012 )andextensionsusingconditional luminosityfunctions( Yangetal. 2012 ; Wangetal. 2013 ).Someoftheseworksalso providetheirownparametricformfortheSHMRasafunctionofhalomass,andwewill usethreeofthemtotourpoints.Thesearetheformsfrom Yangetal. ( 2012 ), Moster etal. ( 2013 )and Behroozietal. ( 2013 )(hereafterY12,M13,B13),whichread: S Y ( m )= M Y 0 m M Y p $ Y + % Y 1+ m M Y p % Y (225) S M ( m )=2 N M 0 m M M p % M + m M M p M 1 1 (226) and log S B ( m )=log( / B M B p )+ f log m M B p # f (0) # log( m ) (227) with f ( x )= # log(1+10 $ B x )+ & B [log(1+exp( x ))] B 1+exp(10 x ) (228) Thesuperscriptlabels { Y M B } refertotheauthornames.Thepositionofthepeakis mostlymodulatedbythepivotmass M Y M B p .InY12andM13,thelowmasslogarithmic slopesaresetby M , Y + Y # 1 andthehigh-massslopesby 0 M , Y # 1 ,respectively.In thecaseofB13,thelinkbetweentheslopesandtheparametersislessstraightforward, butthelowandhigh-massregimesaremostlymodulatedby B and & B 0 B tunesthe high-massbehaviorof log S B ( m ) goingfromlogarithmicat 0 B =0 topower-lawat 0 B =1 (seeBehroozietal.2013).Wesetalloftheseparametersfreewhenperformorthogonal 50

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regressiontsof S Y M B toourmeasurementsoftheSHMR.However,wedoenforce 0 B + 1 and Y + 100 (seeYangetal.2012),whicharelimitsbydenition. Theseauthorsmainlyusemeasuresofthestellarmassfunctionsatdifferent redshiftstobuildaredshiftevolutionmodeloftheSHMR.Theyprovideexplicitredshift dependenceforallparametersinEquations 226 227 .Thus,weusethemtocompare ourmeasurementstothepredictedSHMRoftheseauthorsattheredshiftofoursurvey. ThisisshowninFigure 2-10 ,whereweplottheirpredictionsatlowandhigh-redshift.In addition,thespecicparametervaluesofthe z =1.5 curves,forboththepredictions andthetstoourdata,aredisplayedinTable 2-3 .Thenormalizationvalues N M log / B and log M Y 0 arealsotted,althoughwedisregardanyinterpretationofthembecause thereareimportantsystematicuncertaintiesinthestellarmassesbetweendifferent authors.Athoroughexaminationofthesetoallowameaningfulcomparisonisbeyond thescopeofthiswork.Forthecurrentpurposes,wesimplyassumethatthedifferences instellarmassesareduetoasimplelogarithmicoffset.Thisassumptionholdswell whencomparingdifferentsetsofmassesintheCOSMOSandEGScatalogs.In addition,M13andB13usestellarmassesbasedonBC03andChabrierIMF,which matchesourducialchoiceofmasses.Y12usemassesproducedwiththe Fioc &Rocca-Volmerange ( 1997 )modelsandKroupaIMF,butwestilldonotexpecta signicantdeviationfromaconstantoffsetwhencomparedtoourmasses( Barroetal. 2011b ).Wehavecheckedthisbasedonthemassesfromthisparticularmodelthatare alsoavailableintheEGScontrolcatalog.WealsonotethatweusetheSHMRinY12 thatisbasedonts"CSMF/SMF1",whereonlystellarmassfunctionsareutilized. Welimitthecomparisonbetweenallmeasurementstothecentroidpositionand slopesoftheSHMR.Therearesomediscrepancieswhencomparingourresultstothe predictionsfromtheotherauthors,butthesearenotdramatic(seebelow).Thecentroid oftheSHMRiscomputedastheactualpeakpositionintheparametricrelations,and thetsofthesemodelstoourdatashow log M peak =12.44 0.08 (seeTable 2-3 ). 51

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Wedidnotcomputecondenceintervalsforthe M peak predictionsbecauseitrequires knowledgeoftheexplicitcovariancebetweentheirparametersts.Ourvalueislarger thanthesepredictions;basedonlyonourerrors,itlies0.8 # aboveY12,1.2 # above M13and2.7 # aboveB13.Becausetheirerrorsarenotbeingtakenintoaccount,these offsetsshouldnotbetreatedasabsolutelevelsofinconsistencywithrespecttoour study. Fortheslopes,therearealsosomeslightdiscrepancies.Tobettervisualizethis comparison,wehaveplottedinFigure 2-11 thepredictionandthetstoourdatafor eachparametricmodel.Allcurvesineachpanelarescaledinthex-axistomatchthe peakoftheprediction,andscaledinthey-axistosetallpeakheightstozero.Theidea istoxthepeakposition(inbothaxes)ofallcurvestobettercomparetheslopeson eitherside.Inthiscase,the slopes aretheapproximatepower-lawindexateitherside ofthepeak,andisnotnecessarilylinkedtoaparameterinauniquemanner(except forM13,wheretheslopesareindependentlycontrolledby M 0 M ).Incomparison toM13,ourlowmassslopesaresteeper(higher M )andthehigh-massslopesare shallower(lower 0 M )thantheirpredictions.WithrespecttoB13,thelowmassslopes areinagreementbutourhigh-massslopesareshallower.InthecaseofY12,ourslopes aresteeperatbothlowandhigh-mass. AsexplainedinSection1,thelow-massslopecanbedirectlyrelatedtothe importanceofenergy-versusmomentum-drivenwinds.Ingeneral,wendasteeper low-massslopethanthepredictions,whichfavorsenergy-drivenwinds.Athigh-masses, theinterpretationoftheslopeislessclear,sinceAGNfeedbackandgalaxymergers shouldalsohaveanimportantcontribution. Anotherimportantstudytocompareourmeasurementswithis Wakeetal. ( 2011 ). ItisbasedonHODmodelingof 10 10 M stellarmasslimitedgalaxiesat z 1.5 whichmakesitsimilartoourwork.Theseauthorshadtheadvantageofusingdatawith accuratephotometricredshiftsandstellarmasses,butalsothedrawbackofsampling 52

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Table2-3. Best-tSHMRparameters. Columns1,2: SHMRfunctions(seeEquations 225 227 )andparametersthatdescribethem. Column3: Predictionofthe parametervaluesat z =1.5 ,derivedbytheseauthorsusingdatafrom luminosityandstellarmassfunctionsatdifferentredshifts. Column4: Parametervaluesderivedfromttingthesefunctionstoourclusteringdataat z =1.5 FunctionParameterPrediction z =1.5 SSDFt N M 0.020 0.0070.0139 0.0003 S M log M M p 12.31 0.3212.25 0.02 (Moster M 0.88 0.201.64 0.09 etal.2013) 0 M 0.81 0.120.60 0.02 log M peak 12.3312.44 0.08 log / B -1.70 0.16-2.03 0.17 S B log M B p 11.88 0.1312.03 0.15 (Behroozi B -1.64 0.09-2.10 0.16 etal.2013) 0 B 0.12 0.250.32 0.26 & B 2.65 0.903.31 1.15 log M peak 12.2312.44 0.07 log M Y 0 9.57 0.3110.65 0.13 S Y log M Y p 10.48 0.2210.37 0.30 (Yang Y 0.56 0.110.16 0.04 etal.2012) Y 35. 30100. log M peak 12.3812.44 0.06 asmallregionofthesky(NEWFIRMsurvey,0.25deg 2 ).Theyhadveryfewgalaxies around 10 11 M andthereforeitwasnotpossibletomapthefullpeakoftheSHMR. TheirdataisshowninFigure 2-10 ,wherewehavescaledthestellarmassesby50% toroughlytransformthemfromtheM05modeltoBC03.Theseauthorsperformeda parametrictandfoundapeakat log M peak =12.63 (anestimateduncertaintywasnot provided),whichlies2.5 # aboveourresultof 12.44 0.08 2.7.2EvolutionwithRedshift Atthispoint,wecancompareourresultforthepeakintheSHMRwithotherstudies atdifferentepochsandtraceitsevolutionwithredshift(Figure 2-12 ).WeincludeHOD 53

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11 5 12 0 12 5 13 0 13 5 log M halo 2 2 2 1 2 0 1 9 1 8 1 7 1 6 1 5 1 4 log ( M / M halo ) Behroozi13 z=0 Moster13 z=0 Y ang12 z=0 Behroozi13 z=1.5 Moster13 z=1.5 Y ang12 z=1.5 W ak e11 z 1.5 SSDF z 1.5 Figure2-10. Stellar-to-halomassratiofromourstudyandpredictionsfromother authors.Dashedandsolidlinesarepredictionsat z =0 and z =1.5 respectively.Ourpointsareplottedas log( M lim / M min ) versus log M min .The errorbarsarestronglycorrelatedbetweenneighboringpoints,sinceour galaxysamplesaredenedincumulativemagnitudebins.Wetthe parametrizationsfromthoseauthorstoourdata,robustlymeasuringa maximumat log M peak =12.44 0.08 .Thischaracteristicmassscaleis 4 timeslargerthanwhatisfoundat z =0 .TheM13tisshownasthethin redcurve.Wealsoincludedatafrom Wakeetal. ( 2011 )asemptycircles, wheretheirM05-basedstellarmasseshavebeenincreasedby50%to approximatelymatchtheBC03massesusedbyotherauthors. 54

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12 0 12 5 13 0 0 0 0 1 0 2 0 3 0 4 0 5 log ( M / M halo ) (arbitr ar y offset) M13 12 0 12 5 13 0 log M halo B13 12 0 12 5 13 0 Y12 SSDF Prediction Figure2-11. Comparisonofthehighandlowmassslopesbetweenmodelpredictions { M13,B13,Y12 } at z =1.5 andthetsoftheirparametricmodelstoour data.Ineachpanel,wehaveoffsetallcurvestothesamepeakvalueand shiftedourcurvesinmasstomatchthepeakpositionoftheprediction. Thishasbeendonetohelptheeyeincomparingtheslopesateitherside ofthepeak.Ourdatashowsamoderatediscrepancycomparedtothe predictions. resultsfrom Zehavietal. ( 2011 ), Zhengetal. ( 2007 ), Leauthaudetal. ( 2012 ), Coupon etal. ( 2012 )and Wakeetal. ( 2011 ),aswellaspredictionsfromM13,B13andY12.As mentionedearlier,ourpeakliesabovethepredictionsandbelowthevalueinferredby Wakeetal. ( 2011 ).Lookingatthetrendwithvaluesatotherepochs,thepeakmass seemstohaveevolvedinamonotonicandquasi-linearwaywithredshift.Ourdata supportsachangeoflog M peak =12.44 % 11.8 through z =1.5 % 0 .Thismeansthat thehalomassscalethatismostefcientatformingandaccretingstarstothecentral galaxyhasdecreasedbyafactorof4.5duringthisredshiftrange.Thus,thedownsizing trendofgalaxieshascontinuedsteadilyduringthelast10Gyrs.Lowmassgalaxies havegrownfasterthantheirhaloes,whiletheoppositetrendhappenedforhigh-mass galaxies. 55

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0 4 0 8 1 2 1 6 6 10 14 18 22 M 1 / M min 0 0 0 4 0 8 1 2 1 6 Redshift 11 6 11 8 12 0 12 2 12 4 12 6 12 8 log M p eak Moster13 Behroozi13 Y ang12 Zeha vi11, SDSS Zheng07, DEEP2 W ak e11, NMBS Coupon12, CFHTLS Leauthaud12, COSMOS This study SSDF Figure2-12. Top: Evolutioninthe M 1 / M min ratioforsampleswithdensity n g =10 3 Mpc 3 ,collectedfromdifferentHODstudies.Adeclineinthis ratiowithredshiftismeasuredconsistentlyandagreeswithresultsfrom N -bodysimulations.Thebasicinterpretationisthatathigh-redshiftthereis alargerrateofhaloinfall,whichincreasesthefractionofsimilarmass galaxiesandreduces M 1 / M min (seetext). Bottom: Evolutioninthepeak halomassoftheSHMR.Ourresultsshowthat M peak hasdecreasedbya factorof4.5through z =1.5 # 0 .IncombinationwithotherHOD measurements(points),theevolutionseemstobemonotonic.Thecurves showpredictionsfromCLFandabundancematchingstudies. 56

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2.8SatelliteGalaxies 2.8.1SatelliteFraction ThesatellitefractionsforallofoursamplesaredisplayedonTable 2-1 andplotted inthelowerpanelofFigure 2-13 .Notethatthesatellitesmakingupthisfractionare abovethesampleuxlimit,i.e., f sat doesnotrefertothetotalfractionofsatellitesthata centralgalaxyattheuxlimithas.Thesatellitefractionclearlydecreasestowardsthe brighterend,whichisamanifestationofthedropinthehalomassfunction.Basedupon themodelweuse, M 1 isthescalethatsetstheoccupationnumberofsatellitesinahalo ofagivenmass,andthenumberofsuchhaloesisgivenbythemassfunction.If M 1 approachesthecutoffscaleofthemassfunction,thenthesatellitecontributiontothe totaldensitywillbereducedcomparedtothatofcentralgalaxies.Thiseffectisseenin moststudies( Zhengetal. 2007 ; Zehavietal. 2011 ; Wakeetal. 2011 ; Couponetal. 2012 ; Tinkeretal. 2013 ). Ourfaint-limitvalueis f sat 0.2 (seeFigure 2-13 ).Comparedtotheresultof0.3 obtainedat z =0 ( Zehavietal. 2011 ),itindicatesamildincreaseinthesatellitefraction withcosmictime.Thisconclusionwasalsoreachedby Couponetal. ( 2012 )basedon theircomprehensivestudyofsamplesat 0 < z < 1 .Somesimulationsalsomakesimilar predictions(e.g.,Wetzeletal.2009,2013). 2.8.2The M 1 / M min Relation Adeeperinsightintotherelationshipbetweenhaloesandtheirsatellitesisgivenby the M 1 / M min ratio.AsmentionedinpreviousSections,haloestypicallybecomeoccupied byacentralgalaxyat M min andgainanadditionalsatelliteat M 1 .Thus,atxed M min lowering M 1 woulddirectlyincreasetheoverallsatellitefraction.However, M 1 / M min holdsfurthercluesinrelationtothegalaxiesthatoccupythesehaloes.Becauseofthe declineinthehalomassfunctiontowardsthemassiveend,mostofthehaloesaresmall andhavemassesaround M min .Thesewilltypicallyhostgalaxiesthatarealsosmall, withstellarmassclosetothesamplelimit M lim .Thesatellitesconsideredhavemasses 57

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thatarealsonearthislimitandlivinginhaloesnear M 1 ,wherethecentralgalaxycan haveamassmuchlargerthan M lim .However,if M 1 approaches M min ,thenitscentral galaxywillhaveamasscloserto M lim .Inthecaseof M 1 / M min 1 ,thesatellitewillhave astellarmassaround M lim andthecentralwillbeslightlymoremassivethanthat.Thus, whenthisratioissmaller,thereisanincreasedfractionofcentralsthathaveasatelliteof similarstellarmass. InthelocalUniverse, M 1 / M min 17 ( Zehavietal. 2011 ; Beutleretal. 2013 ). Ontheotherhand,wemeasure M 1 / M min 9 at z =1.5 .Adeclineofthisratiowith redshifthadbeenpredictedbysimulations( Kravtsovetal. 2004 ; Zentneretal. 2005 ) andmeasuredbyabundancematching( Conroyetal. 2006 )andotherHODstudies. Ratiosatdifferentredshiftsandxednumberdensity n g =10 3 Mpc 3 areshowninthe toppanelofFigure 2-12 ,whereitcanbeseenthatthereisageneralincreasetowards latertimes.Thereasonwhyboth f sat and M 1 / M min arehigheratlow-redshiftisdueto theevolutionofthemassfunction.Atlow-redshift,thereareverylargehaloesthatcan hostmultiplesatellites,whichhelpsincreasetheaveragesatellitefraction.However,the fractionofgalaxiesthataresatelliteswithmassesclosetothelimitofthesampleisstill largerathigh-redshift.Thishappensbecausethehaloinfalltimescaleislower,which enhancestheiraccretionrateontootherstructuresandreducesthegapbetween M 1 and M min (seebelowandConroyetal.2006). However,themostinterestingresultwederiveisthetrendof M 1 / M min withsample luminosity.ThemiddlepanelofFigure 2-13 showsindicationsofaslightriseathigh luminosities,whichisnotobviouslyexpected.At z 1 ,thisratiohasbeenobserved tobeconstantordecreasewithincreasingluminosityatxedredshift( Zhengetal. 2007 ; Blakeetal. 2008 ; Abbasetal. 2010 ; Zehavietal. 2011 ; Matsuokaetal. 2011 ; Leauthaudetal. 2012 ; Beutleretal. 2013 ).Simulationsalsopredictthatthe accretionrateislargerformoremassivehaloesatalltimes( Zentneretal. 2005 ; Wetzel etal. 2009 ; McBrideetal. 2009 ; Fakhouri&Ma 2008 ; Fakhourietal. 2010 ),which 58

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wouldlower M 1 / M min atthebrightend.Wemeasuretheoppositetrendat z =1.5 Interestingly, Wakeetal. ( 2011 ),theonlyotherstudyatthisredshiftthatmeasured HODforstellarmassselectedsamples,alsoobtainedaslightincreaseofthisratiowith samplemass.However,thoseauthorsdidnotexplorethiseffectindepth.Theresults from Couponetal. ( 2012 )alsohintasimilartrendat z < 1 inhaloesofmass < 10 13 M althoughtheyareconsistentwithaconstantratio. Inordertobettercomparetheresultsfromafewdifferentauthors,weplot M 1 / M min asafunctionofcumulativenumberdensityinFigure 2-14 .Forvisualclarity,weshow intheleft(right)panelthoseresultsthatfollowanincreasing(decreasing)trendwith density,alongwithourdata.Acaveatinthiscomparisonisthat,inreality,thenumber densityofagivenpopulationdoesnotremainconstantthroughredshift.However, thedatafrom Zhengetal. ( 2007 )and Couponetal. ( 2012 )donotfollowthesame trends,eventhoughtheysamplesimilarredshifts.Thus,fromtheobservationalside,the 0 < z < 1 datadonotofferaconsensusregardingthetrendwithluminosity.At z =1.5 ourresultsandthosefrom Wakeetal. ( 2011 )dosupportaminimalrisein M 1 / M min with luminosity. AsshowninFigure 2-14 ,twofamiliesofcurvescanbedened.Ourresults, Wakeetal. ( 2011 )and Couponetal. ( 2012 )showasimilarshape,offsetinthey-axis accordingtoredshift.Meanwhile, Zehavietal. ( 2011 )and Zhengetal. ( 2007 )aresimilar tooneanother.Onepossibleeffectleadingtothedisparitybetweenthetwosetsof resultsmaybetheparticularselectionofgalaxysamples.Thosein Zehavietal. ( 2011 ), Zhengetal. ( 2007 )and Couponetal. ( 2012 )arelimitedbyabsolutemagnitudeinthe optical. Wakeetal. ( 2011 )selectsdirectlyinstellarmass,andwemakealuminosity selectionthatislatermatchedtoastellarmasslimitedsample.Thus,wendnoclear explanationfortheexistenceofthesetwofamiliesofcurvesregardingsampleselection. Inaddition,alltheseauthors(includingus)useaverysimilarformoftheHOD,and variationsinthechosencosmologydonothavesuchastrongimpact.Regarding 59

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possiblesystematicseffectsinourmodeling,wetestdifferentpossibilitiesinAppendices F and E andndnothingthatwouldalterourconclusions. 2.8.3PhysicalMechanismsforaMass-dependentEvolution Theriseof M 1 / M min withluminosityisnotclearlydetected.However, Zehavietal. ( 2011 )and Zhengetal. ( 2007 )veryclearlymeasuretheoppositebehavioratredshifts z =0 and z =1 ,respectively,sothatevenifourdatafollowsaattrendat z =1.5 ,it wouldimplythatevolutionhastakenplace.Interestingly,thereisnoobviousmechanism thatcouldberesponsibleforthischange,andwespeculatewithsomepossibilitiesin whatfollows.Thedynamicalprocessesatplaycanbereducedtoacompetitionbetween accretionanddestructionofsatellites.Regardingtheformer,bigstructureshave recentlyassembledalargerfractionoftheirmassthansmallercounterparts,atalltimes ( Wechsleretal. 2002 ; Zentneretal. 2005 ; Fakhourietal. 2010 ).Inotherwords,the specicgrowthrateofhaloesisanincreasingfunctionofmass.Regardingthelatter,the dynamicaltimeinbiggerhaloesislarger,contributingtoaslowerdestructionofaccreted satellites.Theseeffectsyieldalargernumberofrecentlyaccretedandundisrupted satellitesinlargerhaloes,whichwouldproduceadecreasein M 1 / M min towardshigher masses. So,whatadditionalmechanismcanreversethistrendathigh-redshift?This mechanismcouldinvolvetheratioofdestructiontoaccretionbeinglargerathigh-masses, whichispossibleifthedynamicaltimescaledecreasesconsiderablywithmass. However,acaveatinthesescenariosisthatweareimplicitlyconsideringthatgalaxies areaccretedordisruptedinthesamewayashaloes,whichdoesnothavetohold. Whatwearereallytrackingaregalaxies,since M 1 / M min isinverselyproportionaltothe occurrenceofgalaxypairswithmassesclosetothesamplelimit.Thus,therecould beastarformationdependentprocessthatdrivesthetrendweseewithstellarmass. Forexample, Wetzeletal. ( 2013 )showthatthestarformationinsatellitesfadesatthe samerateasthecentralgalaxyforafewGyrsafteraccretion,butthenundergoesa 60

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rapidquenchingperiod.Theyndthatquenchingtimescaleisshorterformoremassive satellites.Thus,ifthecentralgalaxyoutgrowsthesatellitesinawayproportionalto itsownmass,thiswouldproducealowerfractionofsimilarmasspairsandplayin favorofourtrend.Inaddition,suchamechanismwouldneedtobecomemildertoward low-redshift,sothatthetrendbecomesinverted.Thisallowsustorestateourprevious question:whatphysicalprocesswouldmoreefcientlyquenchsatellitesinsimilarmass pairsandismoreimportantathighredshift?Wedonothaveaplausibleanswerforthis question. 2.9GalaxyBias Atsmallscales,thecomplexbaryonicprocessesofgalaxyformationbreakthe homologybetweenthespatialdistributionofgalaxiesanddarkmatter.However,at large-scales,thegravitationaleffectsofdarkmatterdominatethedynamicsandthe overdensityofsomeselectionofgalaxiesisexpectedtomatchthatofdarkmatter multipliedbyascalingfactor,thegalaxybias b g .IntheHODmodels,thisquantityis describedasanumber-weightedaverageofthehalobias(seeEquation 220 )and ideallywouldmatchthesquare-rootoftheratiobetweenthelarge-scaleSCFofgalaxies anddarkmatter(Equation 21 ). OurmeasurementsoftheeffectivegalaxybiasareshowninthetoppanelofFigure 2-13 ,whereweplotagainstapparentmagnitudethreshold.Bright(massive)galaxies havealargerbiasthanfaint(small)ones,atrendthathasbeendeterminedinmany otherstudies( Benoistetal. 1996 ; Norbergetal. 2001 ; Tegmarketal. 2004 ; Zehavi etal. 2005 ; Brodwinetal. 2008 ; Brownetal. 2008 ; Foucaudetal. 2010 ; Zehavietal. 2011 ; Matsuokaetal. 2011 ; Couponetal. 2012 ; Beutleretal. 2013 ; Julloetal. 2012 ; Mosteketal. 2013 )andisexpectedbecauseluminousgalaxiesresideonaveragein moremassivehaloes,whicharemorebiasedwithrespecttodarkmatter( Whiteetal. 1987 ; Kauffmannetal. 1997 ).Inadditionto b e # g ,wealsotthelarge-scalebiasdirectly toourmeasuredACFat + > 0.05 deg( 4 comovingMpcat z =1.5 ).Thistdoesnot 61

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Figure2-13. ResultsfromtheHODts.Eachpointdenotesasampledenedbya limitingapparentmagnitudethreshold,whichisassociatedwiththemedian stellarmass M lim .Inthetoppanel,theshadedregionrepresentsthe 1 # intervalofdirectlarge-scalebiasts.TheseareconsistentwiththeHOD bias. 62

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Figure2-14. Ratiobetween M 1 and M min asafunctionofcumulativegalaxynumber density.Wecompareourdata(redpoints)withotherstudies,grouping theminthoseshowingadecrease(leftpanel)oranincrease(rightpanel) withnumberdensity.Thedashedlineindicatestheratioof17presentedin Zehavietal. ( 2011 )asthetypicalvalueforlow-redshiftgalaxies. dependontheHOD,andisperformedbyscalingthedarkmatterSCFinasimilarway totheprocedureinSection 2.5.2 ,butleavingthe z =1.5 biasasafreeparameter.The inputsfromthegalaxypopulationaretheredshiftdistributionsandtheevolutioninthe medianmasstomodulatethebiasacrossredshifts,butnotthegalaxynumberdensity. TheshadedregionsinthetoppanelofFigure 2-13 representthe 1 # condence intervalsforthedirectbiast,whichisingoodagreementwiththeHODvalues.Thus, wendthattheHODmodelingofourdatamakesagooddescriptionofthelarge-scale bias. Nonetheless,wenotethatthisdescriptionisnotperfect.InAppendix F we commentonhowaHODtwithallparametersallowedtovaryfreelymakes # log M oatdowntounphysicalvalues 0 ,tryingtomaintainahighbiasthatotherwisewould 63

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yieldasmallervalueduesmallshiftsinthetted M min and n g .TheoverallHODisnot verysensitiveto # log M andthereforethisisnotasignicantproblem.However,itpoints towardstheamplitudeofourobservedACFsbeingslightlytoolargetobeperfectly reproducedbythecombinationofhalobiasandhalomassfunctions. 2.9.1ComparisontoWakeetal. Wendaslightbiasexcessinourdata,aresultthathasbeennotedtoalarger extentinotherHODstudiesofstellarmasslimitedsamplesat 1 < z < 2 Matsuoka etal. ( 2011 )and Wakeetal. ( 2011 )ndthattheirACFsaretoostrongtobereproduced byahalomodelwiththeobserveddensityofgalaxies.Thosetstotheclusteringplus densitywerecomparedtotstotheclusteringonly,wherethenumberdensitywasnot xedtotheobservedvalue.ThelattertwasabletoreproducetheACFs,butwitha biasabout50%higherthantheclusteringplusdensitytintheirmostmassiveand distantsamples.The z =1.5 samplesof Wakeetal. ( 2011 )aredirectlycomparable toourstudy,sincetheyaredenedbylowerstellarmasslimits.Inthelowerpanelof Figure 2-15 weshowourstandardHODbiasmeasurementsasafunctionofstellar massandthebiasresultsfromthoseauthors.Tomakethecomparisonmoredirect, weplotourresultsfortheMaraston(2005;hereafterM05)evolutionarymodelswith theKroupa(2001)IMF.HerewecommentonthetwotypesofHODtsin Wakeetal. ( 2011 ),andhowtheycomparetoourresults: Clusteringonly : Wakeetal. ( 2011 )ndtheeffectivebiasfromthisttobethe closesttoadirectmeasurementofthelarge-scalebias.However,thesevalues arehighcomparedtoourndings.The0.2dexoffsetrelativetoourworkcould beduetoadifferenceinstellarmassestimates(Figure 2-15 ).Giventhatboth studiesemployverysimilarstellarmassmodels(M05stellargrids,KroupaIMF, andCalzetti2000extinction),thispossibilityseemsunlikely.Anotherexplanation wouldbesamplevarianceduetothesmallsizeofthesurveyin Wakeetal. ( 2011 ), whichcouldleadtoanexcessintheclusteringsignal.Oursurveyisalmost400 timeslargerinareaandthereforesignicantlylessimpactedbythiseffect. Clustering+density : Wakeetal. ( 2011 )ndbiasesfromthistthatarealso notfullyconsistentwithours.Theirbiasislarger(smaller)thanourvaluesinthe low(high)massend.However,theobservednumberdensitiesarediscrepantin 64

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theoppositeway,andthestellarmassatwhichthedensitiesandbiasesmatch isroughlythesame, log M lim " 10.6 .InanyHODmodel,theeffectivebiasis anticorrelatedwithnumberdensityiftherestoftheparametersareheldxed. Thus,if Wakeetal. ( 2011 )andourstudyhadthesameobserveddensities,the HODbiasfrombothsurveyscouldperhapsbeinfullagreement. Thus,wespeculatethatthehighclusteringamplitudein Wakeetal. ( 2011 )mightbe predominantlyaconsequenceofcosmicvariance(asalsosuggestedbythoseauthors), andtheirclustering+densityHODtswouldbeconsistentwithoursiftheobserved z 1.5 comovingnumberdensitieswerethesame. 2.9.2ComparisontoOtherStudies Wealsocompareourbiasresultswithothermeasurementsatdifferentredshifts basedonstellarmassesinthetoppanelofFigure 2-15 .ThesestudiesincludeM10, Foucaudetal. ( 2010 ), Matsuokaetal. ( 2011 ), Julloetal. ( 2012 ), Hartleyetal. ( 2013 ), Mosteketal. ( 2013 )and Beutleretal. ( 2013 ).Thecomparisonsarelessstraightforward thanwith Wakeetal. ( 2011 ),sincetherearesomedifferenceswiththestellarmass modelsusedbyeachauthor.Inaddition,theselectionisnotalwaysdonewithstellar masslowerlimits,butinstellarmassranges.Thus,wechoosetoplotthebiasagainst themedianstellarmassofthefullgalaxysamples.WeshowourresultswithBC03and ChabrierIMFmasses,sincethisisthemostcommonchoiceamongtheotherauthors. Atxedstellarmass,thebiasincreaseswithincreasingredshift.Thisresulthas alsobeenshowninmoststudiesthatusemulti-redshiftdata( Rossetal. 2010 ; Foucaud etal. 2010 ; Mosteretal. 2010 ; Matsuokaetal. 2011 ; Wakeetal. 2011 ; Julloetal. 2012 ; Hartleyetal. 2013 ).Suchbehaviorisexpectedfromanalyticalderivations( Fry 1996 ; Moscardinietal. 1998 )andcanbequalitativelyunderstoodifweassumethat mostofthegalaxiesareformedaroundaparticularredshiftandathighdensitypeaks inthedarkmatterdistribution.Suchgalaxieswouldthenbeinitallyverybiased,butwith timetheirspatialdistributionwouldrelaxtomatchthatofdarkmatter.Thus,thebiasis generallyexpectedtoevolvetowardlowervalues. 65

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10 5 10 7 10 9 11 1 11 3 log M full 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 b e g z 0 z 1 z 1.3 Beutler13 z=0 Matsuoka11 z=0.9 Mostek13 z=0.9 J ullo12 z=1 F oucaud10 z=1.4 Har tle y13 z=1.3 Moster10 z=1.5 SSDF z=1.5 9 8 10 0 10 2 10 4 10 6 10 8 11 0 log M lim 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 b e g W ak e11 Cluster ing+Density z=1.5 W ak e11 Cluster ing z=1.5 SSDF z=1.5 Figure2-15. Top: ComparisonofourHODbiastoothervaluesfromtheliterature,asa functionofstellarmass.Therearesomedifferencesinthewaymasses fromtheotherstudiesaredened,butingeneraltheyrepresentthe medianstellarmassofagivensample.Forthisreason,weshow M full insteadof M lim .WeuseBC03/ChabrierIMFstellarmasses,asdomostof theotherauthors.Wenotetheincreaseofbiasatxedstellarmassasa functionofredshift. Bottom: ComparisonofourHODbiaswith Wakeetal. ( 2011 )asafunctionofstellarmasslimit.Formoredirectcomparison betweentheresults,herewemodelourgalaxieswithM05/KroupaIMF.The clustering-onlytsoftheseauthorsyieldconsiderablylargerbiasvalues thanours,butthismightbeduetosamplevarianceintheirsurvey(see text). 66

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M10donotuseclusteringmeasurements,butanabundance-matchingtechnique basedonthestellarmassfunctions.Theyprovidepredictionsofthegalaxybiasfor severalredshiftsandstellarmassranges.Wehaveplottedaninterpolationofthese valuesinthetoppanelofFigure 2-15 ,takingthemiddlepointoftheirmassrangesas theeffectivemedianmass.Overall,thereisagoodagreementwithourresults. 2.9.3BiasofCentralGalaxies Ideally,onewouldliketopredictthebiasofanindividualgalaxybasedonitsstellar mass.However,thisisnotpossiblebecausethereissomeintrinsicscatter(represented bytheHODfudgeparameter # log M )relatedtootherphysicalprocessesthatmightalso intervene,suchasenvironmentorassemblyhistory.Inaddition,therecanbeensemble scatter,whicharisesifthebiasandmassaredrawnfrompopulationaverages.Thisis whatwehavedonesofarinthiswork,establishingaconnectionbetweentheeffective biasandthemedianmassofagivensample(seeTable 2-1 ),whicharemomentsof broadmassdistributions.Thus,wewishtoreducetheamountofensemblescatterin thebias-stellarmassmapping,whichcanbedonestraightforwardlybyconsideringthe biasofcentralgalaxies.Basically,weexploittheconnectionoutlinedinSection 2.7 wherecentralgalaxieswithstellarmass M lim typicallyoccupyhaloesofmass M min Therefore,thebiasofsuchgalaxiescanbecomputedas b c ( M lim )= b h ( M min ) .Here, thereisnoaveragingoverhalomassesandthestellarmassdistributionislessbroad thanthatofthefullgalaxysamples.Weta4thorderpolynomialtoourresultsandthus provideafunctionalformofthebiasofcentralgalaxiesasafunctionofthestellarmass logarithm m : b c ( m )=1.6+ p 1 ( m # 9.8)+ p 2 ( m # 9.8) 2 + p 3 ( m # 9.8) 3 + p 4 ( m # 9.8) 4 (229) $ p = [ 0.22 0.07,1.38 0.38, # 2.79 0.62,2.23 0.31 ] 67

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whichisvalidovertherange 9.8 < m < 11 2.10Summary Weusearecentlycompleted Spitzer -IRACsurveyover94deg 2 tostudytherelation betweendarkmatterandgalaxiesthroughtheirangulartwo-pointclustering.Ourdata allowsustoselectgalaxiesat z 1.5 withstellarmassesintherange 10 10 # 10 11 M Inordertoderivestellarmassandredshiftdistributions,weemploytheoptical+MIR datafromtheCOSMOSeld( Muzzinetal. 2013b )asareferencecatalog,adaptingitto thephotometryofoursurvey.Then,wedevelopastatisticalmethodthatlinkssources betweentheSSDFandthereferencecatalogbymatchingtheirIRACphotometry, accountingfortherelativephotometricerrorsinbothdatasets.Weareabletoinferwith highcondencethedistributionofstellarmassandredshiftintheSSDFforaparticular IRACselection.IRACmagnitudesandcolorsarewellcorrelatedwiththesequantitiesfor galaxiesintherangeof 1 < z < 2 TheangularcorrelationfunctionsaretwithanHODmodel,whichoffersphysical insightintotherelationshipbetweendarkmatterhaloesandthegalaxiestheyhost,both centralsandsatellites.Ourmainresultsare: Wefullymapthestellar-to-halomassratioacrossitspeak,whichliesinthe middleofthemassrangeweprobe.Thehalomassatthepeakisfoundtobe log M peak =12.44 0.08 .Thisis4.5timeshigherthanwhatisfoundat z 0 supportingthetrendof"archeologicaldownsizing"since z =1.5 .Anevolving quenchingmassscale M q relatedto M peak couldberesponsibleforthiseffect. WecompareourSHMRcurveswiththepredictionsfromotherauthorsat z =1.5 Ourresultsshowahigher M peak than Yangetal. ( 2012 ), Mosteretal. ( 2013 )and Behroozietal. ( 2013 ).Thelow-andhigh-massslopesoftherelationaremore consistentwith Mosteretal. ( 2013 )and Behroozietal. ( 2013 )than Yangetal. ( 2012 ).Inparticular,wemeasureaslightlysteeperlow-massslopethanthese predictions,whichcouldsupportalargecontributionfromenergy-drivenwindsin low-massgalaxies. Theeffectivebiasofgalaxiesisintherange2-4forgalaxiesofstellarmass 10 10 # 10 11 M ,respectively.ThisisingoodagreementwithanHOD-independent tofthelarge-scalebias.Whencomparedtolow-redshiftstudies,wendthatat xedstellarmassthebiasdecreaseswithtime,inagreementwithexpectations 68

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fromtheory.Wealsoprovideattedformofthebiasofcentralgalaxiesasa functionofstellarmass.Thisrelationsufferslessfromensemblescatterthanone thatusesthesampleaverageofthebias, b e # g Thesatellitefractionis 0.2 forgalaxiesofstellarmass M " 10 10 M and decreasestowardthehigh-massend.Incomparisontothehigherfractions 0.3 measuredatlow-redshift( Zehavietal. 2011 ),thisagreeswiththehierarchical CDMscenario,wherewithtimetherearebiggervirializedstructuresthatcanhost multiplesatellites. Wendmildevidenceofanincreaseof M 1 / M min inmoremassivesamples.This isatoddswithwhatisgenerallyfoundatlowerredshifts(e.g.,Zehavietal.2011, Zhengetal.2007)andpredictedbysomesimulations( Wechsleretal. 2002 ; Zentneretal. 2005 ; Wetzeletal. 2009 ).Iftrue,thiseffectimpliesthatat z =1.5 theoverallfractionof M " 10 11 M galaxiesinsimilarmasspairsissmallerthanat lowermasses.Wedonotndaclearreasonforthistrend. Regardingpossiblesystematiceffectsinourtreatment,westressthatourresults arerobust.Ingeneral,wehavenotfoundthatanyofthechoiceswehavemadeabout thettingparametersoroverallHODmodelwouldmakeaqualitativedifferencein ourconclusions.Fixingadifferentnumberofparametersorallowingforandifferential evolutionbetweentheoneandtwohalotermsdoesnotproducesignicantchanges. Thisisinpartduetothestrongconstraintplacedbytheobservedgalaxynumber density,whichisthemaindriverforsetting M min andthebias.Inaddition,iftheprioron thedensityisdropped,theresultsbecomenaturallymorenoisybutstillconsistentwith ourducialmodel.AsexploredinAppendix B ,ourresultsarealsorobustwiththeuseof eitherCOSMOSorEGSdatasetsasthereferencecatalog,eventhoughthephotometry anddataproductsinthosesurveysweregeneratedinverydifferentways.Thisfact stronglysupportstherobustnessofourmethodsandconclusions. Inthenearfuture,deepopticalcatalogsintheSSDFeldwillbeavailablefromthe DarkEnergySurvey.CombiningsuchdatawiththeIRACcatalogsusedinthisstudy willyieldanenormousboosttothiskindofscience.Accuratephotometricredshiftsand stellarmassesforindividualgalaxieswillenableamuchcleanerselection.Suchdata willalsoallowHODmodelingthroughmanyredshiftslicesintherangeof 0 < z < 2 69

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deliveringaconsistentandcomprehensivedescriptionoftheevolutioninthehalo-galaxy connection. Additionally,therearedarkmatterconvergencemapsontheSSDFeldderived fromCMBlensingwiththeSouthPoleTelescope( Carlstrometal. 2011 ).Cross correlationsintheSSDFeldhavealreadybeenperformedby Bleemetal. ( 2012 )and Holderetal. ( 2013 )withanearlyIRACgalaxycatalogand Herschel data,respectively. Thesestudiesfocusedon z 1 sourcesandmeasuredapositivesignal,althoughno halomodelwastted.InChapter 4 ,weuseouraccuratelycalibratedIRACcatalogsto explorethecrosscorrelationof z 1.5 stellarmass-selectedgalaxieswithdarkmatter maps.Thatanalysisoffersadirectconnectionbetweenthesematterelds,andallows foranindependenttestofthehalomodelframework. 70

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CHAPTER3 CROSSCORRELATIONOF Z =1.5 GALAXIESWITHTHECOSMICINFRARED BACKGROUND 3.1Background Whenstarsform,alargeamountoftheuxradiatedintheUV/opticalisabsorbed byinterstellardustandre-emittedinthewavelengthrange1-1000 m.Itisestimated thatabouthalfofthelightthatstarshaveemittedthroughcosmictimehasbeen reprocessedbydust( Fixsenetal. 1998 ; Doleetal. 2006 ),producingtheobserved cosmicinfraredbackground(CIB).Theinfraredluminosityofagivengalaxycanbe usedtodetermineitsspecicstarformationrate(sSFR,Kennicutt1998),whichisa keyquantityinanymodelofgalaxyformation.However,thiscannotbeeasilyapplied toindividualgalaxiesacrossallredshifts.Giventhelargepointspreadfunctions(PSFs) causedbythediffractionlimitofcurrentfacilities,itbecomesverydifculttoresolve distantinfraredsources.Thisisunfortunate,sincemostofthegalaxiesproducing theCIBareat z > 1 ( B etherminetal. 2011 ).Nevertheless,awealthofinformation aboutthestarformationofhigh-redshiftgalaxypopulationscanalsobeunveiledbythe statisticsoftheCIBanisotropies.Theclusteringofgalaxiesisgovernedbythedark matterdistribution,whichallowsonetolinkgalaxiesandtheirpropertiestohalomasses ( Berlind&Weinberg 2002 ; Cooray&Sheth 2002 ; Kravtsovetal. 2004 ).Thus,this methodrepresentsapowerfulstatiscaltooltostudygalaxyformationuptotheepochof reionization. TheeldofCIBclusteringisrelativelynew.Ithasbeenexploredwithobservations intheinfrared-milimeterrangebyseveralauthors( Grossan&Smoot 2007 ; Lagache etal. 2007 ; Vieroetal. 2009 ; Halletal. 2010 ; Amblardetal. 2011 ; Shirokoffetal. 2011 ; PlanckCollaborationetal. 2011 ; Dunkleyetal. 2011 ; Hajianetal. 2012 ; Shangetal. 2012 ; Vieroetal. 2013b ; PlanckCollaborationetal. 2013c ; B ethermin etal. 2013 ),whohaveappliedmodelsofvaryingcomplexitytopreciselydescribethe clusteringoftheCIBanisotropies.Typically,thesemodelsaccountforthedependence 71

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ofIRemissivityonhalomassandtheevolutioninthespectralenergydistributionsofthe dustystarforminggalaxies(DSFGs),andhavebeenabletoplaceconstraintsonstar formationefciencyofhalosandtheevolutionofthecosmicstarformatonratedensity. Thosestudieshaveusedautoandcrosscorrelationsofobservationsindifferent infraredbands.Furtherconstraintsonthehalomodelscanbeplacedbycross-correlating theCIBwithadifferentdarkmattertracer,suchasresolvedgalaxiesselectedatshorter wavelengths.Intheoptical,therearecurrentlynolargeenoughsamplesofgalaxiesthat overlapinredshiftwiththeCIB.However,infraredsourcecatalogsofferaconvenient waytoselectsourcesataredshift z 1.5 ,whichroughlymatchesthepeakoftheCIB redshiftdistribution( Vieroetal. 2013a ; B etherminetal. 2013 ).Inthisstudy,wepresent thecross-correlationbetweengalaxiesselectedwith Spitzer -IRACand Herschel -SPIRE mapsoverthe 90squaredegreeSpitzer-SouthPoleTelescopeDeepFieldSurvey ( Ashbyetal. 2013 ).WesetthehalomodelpropertiesoftheIRACgalaxiestothose foundinChapter 2 .ThisapproachallowsustoplacestrongerconstraintsontheCIB clusteringmodel. ThroughoutthisChapterweusethefollowingcosmology: m =0.27 ! =0.73 H 0 =70kms 1 Mpc 1 and n s =0.96 .. 3.2DatasetsandMaps Forthegalaxypopulation,weusethemainsampledescribedinChapter 2.2 ,which isanearlystellarmass-limitedsampleofgalaxiesat z 1.5 .TheCIBdatacomes fromobservationsmadewiththeSPIREinstrument( Grifnetal. 2003 )intheHerschel telescope( Pilbrattetal. 2010 )at250,350and500 m.Thesewerecarriedoutinthe fast-scanmode,coveringanareaofabout90deg 2 thatalmostfullycoincideswiththe SSDFfootprint.ThepipelineusedtoproducethemapswasSMAP( Levensonetal. 2010 ; Vieroetal. 2013b ),whichoptimizesthealgorithmtoseparatelarge-scalenoise uctuationsfromthesignal.Thetransferfunctionisobtainedbygeneratingsimulated observationsbasedonrealisticestimatesofsignalclustering,andthenprocessed 72

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throughthepipelineaswiththerealobservations.Themapsareconvertedtounits Jysr 1 andhavearounda7%absolutecalibrationerror,whichislowcomparedtothe overalluncertaintiesoftheresultsinthisChapter.ThePSFsare18.1,25.2and36.6 ## for the250,350and500 mbands,respectively. WebinthegalaxyandCIBdataintodensitymapsgivenbythesamegridof 30 ## ) 30 ## pixels,usinganequal-areaWCSprojection( Calabretta&Greisen 2002 ).The galaxysourcecatalogisprojectedasabinnedspatialhistogram,withslightcorrections tothenumbercountsbasedonthesurveycompletenesscalibration(seeAppendix A.3 ), whiletheCIBmapissimplygeneratedonthisgridfromtheSMAPpipeline.Thepixel sizeischosetoapproximatelymatchthelargePSFsofthefar-infrareddata.Wecrop bothdatasetstotheircommonarea,whichisnearlyallofeithersurvey.Furthermore, innerregionsoftheeldweremasked,suchasgapsoflowcoverageintheSSDFand circularaperturesaroundexceptionallybrightobjects.Theseobjectswhereidentied withthenear-infrared2MASSPointSourceCatalog( Skrutskieetal. 2006 )assources with K s ( AB ) < 12 mag.,whicharemostlystars.Thesizeofthecircularmasking apertureswasdeterminedbyanempiricalrelationbetweenthe K s mag.ofasource andthemaximumradiuswhereitaffectedthedetectionoffaintersourcesfromthemain sample.Theareaofthemapsafterthemaskisappliedamountsto85deg 2 .Smoothed versionsofthegalaxyand 350 mmapsareshowninFigure 3-1 ,wheretheclustered powercanbeclearlyseen. 3.3Measuredpowerspectra Themeasuredangularcross-powerspectrabetweenthegalaxyandCIBmaps isshowninFig. 3-2 .Thesespectrahavebeencorrectedwiththetransferfunction oftheSPIREmaps,whichisveryclosetooneformostofthemultipolerange,but becomesverynoisyat l < 300 .Thus,wedecidetosetthelowerboundaryatthatvalue, andtheupperboundaryat l =7000 .Thisisduetotheuncertaintyintheshotnoise, whichbecomesdominantatlargermultipoles.ThisnoiseisPoissonianandarisesfrom 73

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Figure3-1. MapsoftheIRAC-selectedgalaxysampleandtheSPIRE 350 m.Inorder tobettervisualizetheoverdensities,weshowthemsmoothedwitha 2 # kernel. thenitenumberofsourcesinthemapsthatcontributetothesignal.Itisexpected toappearasaconstantpositiveoffset,butitisnoteasytopredictanalytically.An estimationoftherelativeshotnoiseinthedifferentcross-spectraispresentedinSection 3.5 .Animportantadvantageofthistypeofcross-correlationisthatitdoesnotsuffer fromthestronggalacticcirruscomponentthatcontaminatestheCIBauto-spectraat lowmultipoles( Amblardetal. 2011 ; Vieroetal. 2013a ).Instrumentalnoiseisanother componentinthespectra,butwehavecheckedthatitisnegligiblebycomputingspectra withtransposedmaps. Wederiveerrorsfromtwodistinctcomponents.Therstoneiscalculatedfromthe errorsinthetransferfunctionandimportstheuncertaintiesregardingthemap-making. Thesecondstemsfromjackknifesamplingandisabletorecovertheerrorfromsample variance.Wedeterminethecovariancematricesbetweenthe C l valuesfromeach ofthesecomponents, C TF and C JK ,respectively.TheerrorbarsshowninFig. 3-2 representthesuminquadratureofthediagonalelementsfrombothmatrices. 74

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10 3 10 4 l 10 2 10 1 C l (Jy sr 1 ) 250 m 10 3 10 4 l 10 2 10 1 350 m 10 3 10 4 l 10 2 10 1 500 m Figure3-2. Angularpowerspectrabetween z 1.5 IRAC-selectedgalaxiesandthe SPIREmapsovertheSSDFregion.Thesolidlinesrepresentthebest-tby ourmodel,whichusesthedatainall3bandssimultanouslytoparametrizea redshift-dependenthalomodelofthefar-infraredluminosityofstar-forming galaxies.ThesecurvescontainthesignalfromgalaxiesplusanPoissonian noiseoffset,whichisalsoafreeparameterinthemodel.Thesignalisbuilt byaddingthepowerspectrumcomponentsfromone-andtwo-halogalaxy associations.Theone-halotermandtheconstantnoiselevelare responsiblefortheupturnathighmultipoles. 3.4Theory 3.4.1Two-pointSpectrum Theangularcross-powerspectrumoftheuctuationsintheangulargalaxy overdensity g ( andCIBspectraluxdensities I # atanobservedfrequency 1 isdened as: & g ( lm & I l m # = C l g # & ll & mm (31) I # resultsfromthecumulativeemissionofsourcesfollowingaredshiftdistribution,and thuscanbelinkedtothecomovingemissivitydensity j # as j # ( z )=(1+ z ) dz d dI # dz ( z ). (32) 75

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Then,thethree-dimensionalpowerspectrum P g # ( k z ) betweentheemissivitiesandthe comovinggalaxyoverdensity g s canbedenedas & g s ( k z ) & j # ( k # z ) =(2 2 ) 3 j # ( z ) P g # ( k z ) & 3 ( k # k # ), (33) where j # representstheaverageatthegivenredshift.Theconnectionbetween Equations( 31 )and( 33 )canbedescribedthroughtheLimberprojection( Limber 1953 ),whichintheat-angleapproximationreadsas C g # l = $ & 0 dz H ( z ) c 2 ( z ) W # ( z ) W g ( z ) P g # ( k = l / ( z ), z ), (34) where H ( z ) istheHubblefunction, ( z ) isthecomovingdistanceandtheredshift distributionsofthetracersare W # ( z )= dI # ( z ) dz (35) whichhasunitsofJysr 1 ,and W g ( z )= dN g ( z ) dz 1 dz # dN g ( z ) dz (36) where N g isthetotalnumberofobservedgalaxies.Notethat W g isdimensionlesssince thegalaxymaphasbeenconvertedtooverdensities. 3.4.2Halomodel Tomodelthe3-Dpowerspectrum,wewillrstdenehowwelinktheCIBemission todarkmatterhalos,buildingamodelsimilarto Shangetal. ( 2012 ).Allmassintegrals thatfollowarecomputedintherangeof 10 8 # 10 16 M Theaverageemissivitycanbeexpressedas j # ( z )= & dM dN dM ( z ) 2 f cen # ( M z )+ f sat # ( M z ) 3 (37) 76

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Here, dN / dM isthehalomassfunctionfrom Tinkeretal. ( 2010 )and( f cen # f sat # )arethe luminosity-weightednumberofcentralandsatellitegalaxiesinahaloofmass M .These taketheform f cen # ( M z )= L (1+ z ) # ( M z ) (38) and f sat # ( M z )= & dm dn dm ( m z | M ) L (1+ z ) # ( M z ). (39) Thefunction dn / dm isthesubhalomassfunctionforsubhalosofmass m residingina hosthaloofmass M .Weadopttheparametrizationof Tinker&Wetzel ( 2010 ). L # ( M z ) isthe(sub)haloluminosity,whichweassumetobethesameforhalosandsubhalosof thesamemass,andwillbedescribedinthenextsubsection. Thepowerspectrumcanbedecomposedinto1-haloand2-haloterms: P g # ( k z )= P 1 h g # ( k z )+ P 2 h g # ( k z ), (310) with P 2 h g # ( k z )= 1 j # ( z ) D # ( z ) b g ( z ) P lin ( k z ), (311) where P lin ( k z ) isthelinearmatterpowerspectrumand b g ( z ) istheredshift-dependent galaxybias,whichwextotheformfoundinChapter 2 .Theluminosity-weightedCIB biastakestheform D # ( z )= & dM dN dM ( z ) b h ( M z ) 2 f cen # ( M z )+ f sat # ( M z ) 3 (312) where b h ( M z ) isthehalobiasfrom Tinkeretal. ( 2005 ).The1-halotermismodeledas 77

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P 1 h g # ( k z )= 1 j # ( z ) n g ( z ) & dM dN dM ( z ) ) 2 f cen # ( M z ) N sat g ( M z ) u ( k M z )+ f sat # ( M z ) N cen g ( M z ) u ( k M z )+ f sat # ( M z ) N sat g ( M z ) u 2 ( k M z )]. (313) Thefunction u ( k M z ) istheFouriertransformoftheNFWproleofahalo( Navarro etal. 1997 ; Cooray&Sheth 2002 )and N cen g ( M z ) N sat g ( M z ) arethenumberof centralandsatellitegalaxiesasafunctionofhalomassandredshift.Thecomoving galaxydensityisrepresentedby n g ( z ) .Thesequantitiesrelatetothehalooccupationof galaxies,andweredeterminedat z =1.5 inChapter 2 .Here,weneglecttheirredshift evolutionandxthevaluestothosefromChapter 2 .Thisapproximationisvalidsince theredshiftdistributionofgalaxiesisstronglypeakedaround z =1.5 ,anddeviations fromthishalosolutionacrossthegalaxysamplewouldaverageout(seeChapter 2.3.2 ). Thus,weset n g =1.9 ) 10 3 Mpc 3 and N cen g ( M )= 1 2 1+erf log M # 11.96 0.2 ./ (314) N sat g ( M )= N cen g ( M ) M # 1.2 ) 10 12 M 7 ) 10 12 M . (315) Wehavecheckedthattheuncertaintiesintheparameterscontrollingtheseexpressions haveanegligibleeffectintheresultsofthiswork,andthereforewenditunnecessary topropagatethem. AnotherimportantobservablethatthismodelcanpredictistheaverageSFR densityoftheUniverseasafunctionofredshift.Itrelatestotheemissivityas( Planck Collaborationetal. 2013c ) 78

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3 SFR ( z )= K j # ( z ) (1+ z ) 2 ( z ) s e # # ( z ) (316) where K =1.73 ) 10 10 M yr 1 L 1 ( Kennicutt 1998 ),and s e # # ( z ) istheaverage observer's-frameSEDatagivenredshift,normalizedto1 L over 8 # 1000 m.The shapeofthisSEDwillbeintroducedinthenextsubsection. NotethatEquation( 316 )accountsonlyforthedust-obscuredstarformation, whichishoweverdominantovertheunobscuredtypeat z 1 (seeFig.11in Planck Collaborationetal. 2013c ). 3.4.3Haloluminosity Wedecomposethehaloluminosityintodifferentparts: L # ( M z )= L 0 # ( M z ) $( z 1 ) % ( z ). (317) L 0 isanormalizationfactorand # ( M z ) istheluminosity-halomassrelation.This functioncanbeparametrizedasalog-normaldistribution( Shangetal. 2012 ): # ( M z )= M + 2 2# 2 LM exp # (log M # log M p ( z )) 2 2 # 2 LM / (318) Here, log M p ( z ) representsthehalomasswithhighestspecicemissivity(andSFR),and weparametrizeitas log M p ( z )=11.7 ) (1+ z ) q (319) Theredshiftevolutionofthischaracteristicmassisintroducedafterthegrowing evidencethatstarformationatxedhalomasswasmoreefcientinthepast( Moster etal. 2013 ; Behroozietal. 2013 ; Yangetal. 2012 ),andhasbeenfoundtobe log M p 11.7 at z =0 ( Zehavietal. 2011 ; Behroozietal. 2013 ). TheshapeofaverageinfraredSEDatagivenredshiftisrepresentedby $ ( z 1 ) WemakeuseoftheSEDfunctionalformsofmain-sequenceDSFGspresentedin 79

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B etherminetal. ( 2013 ),whicharebasedonthelibrariesfrom Magdisetal. ( 2012 ). ThesespectraaccountforacarefultreatmentofdifferentcomponentsoftheIRemission ("warm"diffuseinterstellarmediumand"hot"photo-dissociationregions)aswellasthe riseofdusttemperaturewithredshift.TherelativescalingofthefullSEDatdifferent redshiftsisgovernedby % ( z )=(1+ z ) ) (320) whereapositivevalueof 4 canaccountfortherisingstarformationactivityatearlier epochs,atrendthathasbeenpointedoutinmanystudies(e.g., Noeskeetal. 2007 ; Elbazetal. 2007 ). 3.5Fittingapproach BesidestheparametersthatdenethemodelfromSection 3.4.2 ,weintroduce anotherquantitythatmodulatestheunknownamplitudeofthePoissonnoiseinthe powerspectra, K noise .AsmentionedinSection 3.3 ,thisnoisestemsfromthefactthe observedIRuxandthegalaxydensityareproducedbyanitenumberofobjects. ThenoiselevelinthecrossspectrafromFigure 3-2 isdifferentineachbanddueto thedifferentemissivities,buttheCIBsourcesthatproducethemarethesame.Forthis reason,wechooseonly1parametertomodulatethenoise,whichisthenscaledfor eachbandaccordingtotheoverlapbetween W # ( z ) and W g ( z ) .Theabsolutenoise levelsareestimatedas C noise # = K noise & dz W # ( z ) W g ( z ) (321) Then,wedenethesetoffreeparameterstotas P = { log L 0 q # 2 LM 4 ,log K noise } Thettingapproachisaimedtomaximimzethefollowinglikelihoodoftheobserveddata vector C obs l tobearealization C mod l ( p ) ofamodeldenedbytheparameters p ,including constraintsfromalistofpriors : 80

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ln L ( p | C obs l ) $ # % i =TF,JK [ C obs l # C mod l ( p )] C 1 i [ C obs l # C mod l ( p )] T # % j 0 j # 0 mod j ( p ) 2 2 # 2 j (322) Weimposetwosetsofpriors.TherstoneisthelocalSFRdensity 3 SFR ( z =0)= 2.1 ) 0.4 ) 10 2 M yr 1 Mpc 3 from Vaccarietal. ( 2010 ),transformedaccordingto our H 0 .ThesecondsetarethetotalCIBintensitiesmeasuredby Lagacheetal. ( 2000 ): 1 I # =10.4 2.3,6.5 1.6,2.6 0.6nWm 2 sr 1 forthe250,350and500 mbands, respectively. 3.6Resultsanddiscussion WeperformaMonteCarloMarkovChainanalysisoftheparameterspaceusing theMCHammercode( Foreman-Mackeyetal. 2013 ).Themarginalizedlikelihoodsare showninFigure 3-3 andthebest-tparametervaluesaredisplayedinTable 3-1 ,witha 2 best =22.2 for41degreesoffreedom( 3 ) 14 l -bins,4priors,5freeparameters). Since L 0 modulatesanarbitrarynormalizationand K noise simplytracesthePoisson noiselevel,theyarenotrelevantfortheinterpretationofourCIBhalomodel.Thus,we willfocusthediscussionontheremaining3parameters.Thepositivevalueof q implies thatthedatafavorshigherstarformationefcienciesathighredshift. Vieroetal. ( 2013a ) and PlanckCollaborationetal. ( 2013c )useasimilarhalomodeltotSPIREpower spectra,butwithanon-evolving M p .Theyndvalues log M p 12.5 ,whichwouldalso supporttheevolutionarytrendifourconstraintof log M p ( z =0)=11.7 isenforced.We show log M p ( z ) inFigure 3-4 ,alongdatafromotherauthors. Thestate-of-the-artstudyontherelationbetweentheCIB,galaxiesandhalosis thatof B etherminetal. ( 2013 ),whouseacomprehensivephysicalmodelthatlinks well-calibratedsSFRingalaxiesfromthemid-infraredtoradiowavelengths.Ourresults 81

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inFigure 3-4 predictastrongerevolutionin M p ,althoughstillconsistentgiventheerrors. Wealsoshowthemeasurementofthestellar-to-halomasspeakat z =1.5 from Chapter 2 .Althoughnotexactlythesamequantityas M p ,itiscloselyrelated.Their valueisverysimilartoourpredictionforthesameredshift. Regardingthescatterintheluminosity-halomassrealtion, # 2 LM ,ourresultisin broadagreementwith Vieroetal. ( 2013a )and PlanckCollaborationetal. ( 2013c ). However,thisvalueisdifculttointerpret,giventhesimplicityoftheformof # ( M z ) weuse.Therealshapeofthisfunctionmaybemorecomplexandnotfalloffassteeply ( Behroozietal. 2013 ; Mosteretal. 2013 ),diminishingtheconceptualsignicanceofan equivalentGaussianvarianceasusedhere.Moreover,iftherealrelationhashigh-or low-massslopesthataresignicantlyatter,thenthiscouldbiasthe M p valuemeasured here.Forinstance, B etherminetal. ( 2013 )showsaatterslopeathighmasses,which wouldpointtoour M p beingoverestimated. TheredshiftdistributionsoftheobservedCIBuxasconstrainedbyourdataare showninFigure 3-5 .Theredcurvesrepresent W # ( z ) transformedtounitsofpower. ThetoppaneldisplaystheredshiftdistributionofgalaxiesasderivedinChapter 2 ,which stronglyoverlapswiththeCIBdistributions.Thisoverlapisresponsibleforthehigh signal-to-noiseratiointhemeasuredpowerspectrafromFigure 3-2 .Theemissivities arehigherthanthosefrom B etherminetal. ( 2013 ),whichisareectionofthehighvalue of 4 .Thisparameterrepresentshowsteeplythecosmicstarformationrateincreases towardhigherredshifts,andishighcomparedtothevalues 2.5 0.1 and 3.6 0.2 found by Vieroetal. ( 2013a )and PlanckCollaborationetal. ( 2013c ),respectively. Finally,withtheuseofEquation( 316 )wecanderivetheaveragestarformation ratedensityoftheUniverseasafunctionofredshift.ThisisshowninFigure 3-6 .The peakinstarformationactivityoccursat z 2 ,inagreementwithexpectations(e.g., Hopkins&Beacom 2006 ; Gonz alezetal. 2010 ; B etherminetal. 2013 ).However,we measurearelativelystrongeramplitudeat z > 1 ,whichisdrivenbythelargevalueof 4 82

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Table3-1. Best-tCIBhalomodelparameters. log L 0 q # 2 LM 4 log K noise # 5.4 +0.08 0.5 0.08 +0.03 0.02 0.5 +0.6 0.2 3.9 +0.7 0.3 # 2.9 +0.1 0.3 Figure3-3. MarginalizedposteriorlikelihoodsfortheparametersinourCIBhalomodel. Thegreenandredpointsrepresentthelocationsofthepeakintheposterior distributionsandthelocationoftheminimumchi-squaredfound, respectively.Thecontoursrepresenttheregionsencompassing68.3%, 95.5%and99.7%oftheposteriordistributions. 3.7Summary Wehaveperformedaparametrictofahalomodeltothecross-powerspectrum betweenthe250,350and500 m Herschel /SPIREbandsandasourcecatalogof Spitzer /IRAC-selectedgalaxiesat z 1.5 .Themeasuredspectrahaveanexcellent signal-to-noiseratio,owingto(1)thestrongoverlapintheredshiftdistributionsof thesignalinthesedatasets,(2)theadvantagethatthistypeofcross-correlationis 83

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Figure3-4. RedshiftevolutionofthehalomasswherethespecicIRluminosityis highest.Theredlineshowsthebesttfromourmodelalongwiththe 1 # contoursastheshadedregions.Thediamondpointisthemeasurement derivedinChapter 2 ofthepeakmassinthestellar-to-halomassrelation, whichiscloselyrelatedto M p .The M p evolutionfromourworkissteeper thanthatpredictedby B etherminetal. ( 2013 ).Apossiblereasonforthis discrepancycouldtheoverestimationofour M p valuesasaconsequenceof thesimpliedfunctionalformweusefortheluminosity-halomassrelation (seetext). relativelyinsensitivetospurioussignals(suchasgalacticcirrae)whichareaconcernin auto-correlationsand(3)thelargeareacovered( 90 sq.deg 2 )overacontiguouseld, allowingustoprobeaverylargecomovingvolume. WexthehalomodelpropertiesofthegalaxysampletothatfoundinChapter 2 ,inordertoisolatethedegreesoffreedomthatdescribetheCIBevolutionthrough 0 < z < 4 .Weuseasimplemodelbasedon Shangetal. ( 2012 ),butwith2main 84

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0 5 1 0 1 5 2 0 2 5 3 0 3 5 2 4 6 8 10 5 dN gal /dz 0 5 1 0 1 5 2 0 2 5 3 0 3 5 1 3 5 250 m This study Bether min13 0 5 1 0 1 5 2 0 2 5 3 0 3 5 1 2 3 d ( I ) /dz (nW / m 2 / sr) 350 m 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 Redshift 0 0 0 5 1 0 500 m Figure3-5. Thetoppanelshowstheredshiftdistributionofthetotalnumberofsources inthegalaxysample.Thelowerpanelsshowinredthebest-tdistributionof theobserveduxspectraldensity I # forthe3bands,convertedtounitsof power.Thedashedlinerepresentsthepredictionsfrom B etherminetal. ( 2013 ).Notethat W g ( z ) and W # ( z ) areproportionaltothecurvesinthetop andlowerpanels,respectively. 85

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0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 Redshift 0 00 0 05 0 10 0 15 0 20 0 25 SFR ( M yr 1 Mp c 3 ) This study Bether min13 Figure3-6. Evolutionoftheobscuredstarformationratedensity,obtainedfromtheIR emissivitythroughtheconversionfactorin Kennicutt ( 1998 ). differences.First,weallowaredshiftevolutioninthecharacteristichalomasswhere thespecicIRluminosityishighest.Second,weuseSEDlibrariesfrom B ethermin etal. ( 2013 )and Magdisetal. ( 2012 )tomodeltheemissivitiesasafunctionofredshift, insteadofadoptingamodiedblackbodyspectrum.Webelievethischoicemakesa signicantimprovementtowardsamorerealisticmodel. Ourresultspointtowardsasteepincreaseinthecosmicstarformationdensity towardshigherredshifts,aswellasinthehalomasswherestarformationismost efcient.Althoughthesetrendshavebeenwellestablishedinrecentyears,ourndings supportaparticularlystrongevolution.However,wewarnthatourmodelcontainssome simplicationsthatmightbiastheresults,especiallyregardingtheparametricformofthe luminosity-halorelation.Amoredetailedformwillbebettermodeledandconstrainedin 86

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thenearfuturewiththereleaseofphotometricredshiftcatalogsfromtheDarkEnergy Survey,whichwillallowforanetomographiccross-correlationacrossredshift. 87

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CHAPTER4 CROSSCORRELATIONOF Z =1.5 GALAXIESWITHCMBLENSING 4.1Background TheobservedmapofanisotropiesfromtheCosmicMicrowaveBackground(CMB) containsarcminute-scaledistortionsduetogravitationallensingbymatterstructurein theUniverse.PhotonsemittedfromtheCMBtravelthroughaninhomogeneusdensity distribution,wherelocalgradientsinthegravitationalpotentialcausedeviationsinthe travelingdirection.Thisresultsinanintegrateddeectionangle $ d forraysobserved alongagivenlineofsight $ n .Thisallowsonetodeneascalarlensingmapintermsof theconvergence ) ( $ n )= #/ $ d ( $ n ) / 2 ,whichrepresentstheimagemagnicationandis proportionaltotheintegratedmatterdistribution.Inpractice,theapproachtoderive ) ( $ n ) istomeasurethecouplingthatlensinginducesbetweenFouriermodesontheCMB temperaturemap,whichotherwisewouldbeuncorrelated.Generally,thecovariance matrixofthesemodesisusedinthequadraticestimatorfrom Okamoto&Hu ( 2003 )to ndthemost-likelydistributionofthelensingpotential. ThemainapplicationofCMBlensinghasbeencosmological,withconstraints onthegrowthofstructureanddarkenergy( Sherwinetal. 2011 ),neutrinomasses ( vanEngelenetal. 2012 ; PlanckCollaborationetal. 2013a )andpossiblytheenergy scaleofinationthroughpolarizationmodes( Kamionkowskietal. 1997 ; Hansonetal. 2013 ).However,CMBlensingcanalsoshedlightontheassociationofthedarkmatter andgalaxypopulations.Severalstudieshavefollowedthisapproachtodetermine howstronglydogalaxysamplestracethelargescalematterstrucuture.Suchstudies haveusedquasars( Sherwinetal. 2012 ; Geachetal. 2013 ),luminosity-selected galaxies( Bleemetal. 2012 ; PlanckCollaborationetal. 2013a )andthecosmicinfrared background( Holderetal. 2013 ; PlanckCollaborationetal. 2013b ),ndinginallcases ahigh-signicanceclusteringwithrespecttotheCMBlensingmap. 88

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Thestrengthofthematter-tracercorrelationisusuallyquantiedwiththe bias ( Peebles 1980 ),whichrepresentstheaverageratiobetweengalaxyanddarkmatter overdensitieswithinsomespatialscaleandforagivencosmology.Inthecurrent paradigmofstructureformation,baryonsanddarkmatterfollowacloselyhomologous distributionatlargescales.Thisimpliesthatthegalaxypowerspectrumshouldequal thatofdarkmattertimesascalinggivenuniquelybythegalaxybias.Thebiasobtained fromthegalaxy-galaxycorrelation, b gg ,shouldbethesameastheoneobtainedfrom thegalaxy-lensingcorrelation, b g .Thus,thecomparisonofthesequantitiesrepresents anrevelatorytestofthedataandthemodelsused,includingcosmologicalassumptions. Sherwinetal. ( 2012 )crosscorrelatedCMBlensingmapsfromtheAtacama CosmologyTelescope( Swetzetal. 2011 )withahigh-redshiftquasarsamplefrom SSDS,ndinganexcellentconsistencywiththebiasvaluederivedfromthequasar auto-correlation.InthisChapter,weperformanother b gg # b g test,employingdifferent datasetsinadifferentregionoftheskythan Sherwinetal. ( 2012 ).Weusea z =1.5 Spitzer /IRAC-selectedsampleofgalaxies( Ashbyetal. 2013 )andaCMBconvergence mapfromtheSouthPoleTelescope( Carlstrometal. 2011 ).The b gg valuesareadopted fromChapter 2 ,whereananalysisofthegalaxy-galaxyclusteringandahalomodel interpretationarepresented. ThroughoutthisChapterweusethefollowingcosmology(following vanEngelen etal. 2012 ): b h 2 =0.02235 c h 2 =0.1086 H 0 =70.92kms 1 Mpc 1 =0.0878 n s =0.96 and A s =2.453 ) 10 9 .Weusethedenitionofthelasttwoquantitiesatthe referencewavenumber k 0 =0.002Mpc 1 .Thiscosmologyproduces # 8 =0.79 4.2Datasets WeusethemaingalaxysamplefromChapter 2 ,whichhasaredshiftdistribution peakingat z 1.5 .Forthelensingmaps,weusedataderivedfromtheTheSouthPole Telescope( Carlstrometal. 2011 ).Thisfacilityhasimaged2500deg 2 at150GHzdown toaprecisionof18 K .Thisareaisdistributedamongseveralelds,andweusedata 89

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fromthe 100 deg 2 centeredat(23h30m,-55d),whichwastakenduringthe2010-2011 observingseasons.WefocusonthiseldbecauseitiswhereSpitzerdataisalso available.Weuselensingconvergencemapspresentedin vanEngelenetal. ( 2012 ). Thesemapswerederivedusingthequadraticestimatorfrom Okamoto&Hu ( 2003 ) andhavebeencleanedofpointsourcesandthermalSunyaev-Zeldovichdetections withsignal-to-noiseabove6.Moreover,foregroundcontaminationinthetemperature mapsdoesnotproduceasignicantsystematiceffectinlensingreconstructionbeyond afewpercent( vanEngelenetal. 2012 ).Wealsousesimulatedconvergencemapsthat includearealisticcontributionfromvariouscontaminants,renderingagoodestimation ofthenoise.Wewillusethesesimulatedmapstoderiveerrorsinourresults.In addition,thelteringappliedtotheSPTdatamakesitneccessarytocorrectthepower spectraofthelensingmapwithatransferfunction.Thiswasderivedthroughthecross correlationbetweenthelensingpotentialspectrumusedtomakeofthesimulatedmaps andthespectrumofthesesimulationsrecoveredafterthemap-making. WeuseconvergencemapsderivedfromSPTpoldata,apolarizationinstrument attheSouthPoleTelescope( Carlstrometal. 2011 ).Theinstrumentcontains polarization-sensitivebolometersat95and150GHz.Theobservationsweretaken overanareaof100deg 2 thatcoversthefullSSDFeld,overtwoseasonsin2012 and2013.TheimagesineachbandweredecomposedintotheI,QandUStokes parametersinordertoquantifythelevelofpolarization.Finalnoiselevelsare25and 10 Karcmin 1 forthe95and150GHzbands,respectively.Detailsontheinstrument andobservationscanbefoundin Austermannetal. ( 2012 ), Georgeetal. ( 2012 ), Sayre etal. ( 2012 )and Henningetal. ( 2012 ).Areviewispresentedin Hansonetal. ( 2013 ). 4.3Maps Galaxyandlensingmapswherebinnedintomapswith 2 # ) 2 # pixels,inorderto approximatelymatchtheresolutionfromthelensing.Thegalaxycountsmapwasthen transformedintoandover-densitymap.Moreover,aeldmaskwasgeneratedusing 90

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Figure4-1. Fromlefttoright:SSDFgalaxyoverdensitymap,SPTconvergencemap, andgalaxyoverdensitywithconvergencecontours.Thesemapshavebeen maskedforbrightsourcesasoutlinedinSection 4.3 andsmoothedwitha 18 ## gaussiankerneltoenhancethevisualizationoflargescalemodes.Red colorsrepresenthigherdensity/magnication.Itcanbeclearlyseenhow manyoftheconvergencepeaksandthroughsmatchthegalaxydistribution, whichproducesthecorrelatedsignalinthecross-spectrum. thecontoursfromtheSSDF.Brightstarsandlowcoveragegapswerealsomasked, followingtheprocedurefromChapter 2.2 Figure 4-1 showsthemaskedgalaxyandconvergencemaps,whichhavebeen smoothedwithan 18 ## gaussiankernelinordertobettervisualizethespatialuctations atlargescales,wheremostofthecross-powerwillcomefrom.Intheright-mostpanel weshowtheconvergencecontoursoverthegalaxydensitymap.Thecorrespondence ofpeaksanddepressionsisvisuallyapparent. 4.4Theory Atagivenlineofsightvector $ n ,theCMBlensingconvergence ) iscalculatedasthe integralofthematteroverdensity & uptothesurfaceoflastscattering: ) ( $ n )= z cmb & 0 dz W ( z ) & ( ( z ) $ n ), (41) where ( z ) isthecomovingdistanceand W ( z ) isthedimensionlesslensingkernel givenby W ( z )= 3 H 2 0 2 H ( z ) c m (1+ z ) ( z ) cmb # ( z ) cmb (42) 91

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Here, cmb isthecomovingdistancetothesurfaceoflastscattering( z cmb 1100 )and H ( z ) istheHubblefunction.Forgalaxies,wecandenearedshiftkernelbasedontheir distribution dN / dz W g ( z )= dN ( z ) dz 1 dz # dN ( z ) dz (43) TheseredshiftkernelsareshowninFigure 4-2 .Althoughtheredshiftdistributionof theconvergenceisratherbroad,itplateausataverysimilarredshifttothegalaxies, boostingthecross-correlationsignal. Finally,theangularcrosspowerspectrumbetweengalaxiesandlensingconvergence atagivenmultipole l canbederivedas C g l = $ & 0 dz H ( z ) c 2 ( z ) W ( z ) W g ( z ) b ( z ) P ( k = l / ( z ), z ), (44) where b isthegalaxybiaswithrespecttodarkmatterand P ( k z ) isthelinearmatter powerspectrumatagivenredshift( Lewis&Challinor 2006 ). 4.5Cross-correlation Thegalaxy-galaxyandgalaxy-convergencespectraareshowninFigure 4-3 Welimitour l -rangetomultipolesunder 1000 becausethelensingsignalbecomes toonoisybeyondthatpoint,andalsobecausewewanttofocusonscalesthatare inthelinearregime.Athigher l ,theone-haloterminthegalaxy-galaxyspectrum becomesadominantsourcesofpower.Inthecross-spectrum,however,wedonot expecttheretobeasignicantcontributionfromaone-haloterm,giventhelimited resolutionofthelensingmap.Thenoiseinthismapdominatestheuncertaintyof thecross-spectrum,andwederiveerrorbarsbycross-correlatingthegalaxymap withsimulatedlensingmaps,whichareuncorrelated.Forthegalaxies,errorbarsare obtainedfromajackknifesampleoftheeld.Moreover,weestimatethePoissonwhite noisefromthegalaxy-galaxyspectrumas P gg noise = s / N gal =4.7 ) 10 8 ,i.e.theratio betweenthesolidangleofthesurveyandthetotalnumberofgalaxiesinthemap. 92

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Figure4-2. Top:Redshiftdistributionofthegalaxyandconvergencekernels, normalizedarbitrarilyfordisplaypurposes.Thereisasignicantoverlap betweenthesedistributions.Bottom:Galaxybiasasafunctionofredshift, constrainedfromthegalaxyauto-correlationinChapter 2 Weadoptthegalaxy-galaxylargescalebiasatthepivotredshift z =1.5 from Chapter 2 ,where b gg =2.2 0.1 ismeasured.Wedonotperformattothe galaxy-galaxyspectrum,whichiscomputedinthisworkjustfordisplaypurposes. ThepredictedpowerspectrumusingthisbiasvalueisshowninthetoppanelofFigure 4-3 .ThissignalincludesthePoissonnoiseoffset,butnottheone-halotermcomponent. Forthisreason,thecurveliesbelowthehighl points.Westressthatlargescalebias 93

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valueiscorrect,whichinChapter 2 wasfoundtobeconsistentwiththeeffectivegalaxy biasfromathoroughone-andtwo-halotreatmentofhalooccupationdistributions. Inthiswork,wetthegalaxybiastothegalaxy-convergencespectrum.Thetting procedureisequivalenttothatinChapter 2 ,whichconsistsofminimizingagaussian likelihooddenedby b g astheonlyfreeparameter.Theredshiftevolutionofthebias isdeterminedineachrealizationbyscalingthisvalueaccordingtotheprescriptions fromChapter 2 ,which,inessence,renormalizethe b ( z ) from Mosteretal. ( 2011 )to match b ( z =1.5)= b g .Weobtain b g =1.3 0.3 ,avaluethatisinconsistentwith b gg =2.2 0.1 atthe 3 # level. 4.6Discussion Wehavefoundaverylargediscrepancy( 3 # )inthegalaxybiasasmeasured fromthegalaxy-galaxyandgalaxy-convergencespectra.Whilethisissurprising,it issupportedbyadditionalevidencein B etherminetal. ( 2013 ).Theseauthorsbuilda redshift-dependenthalomodelofthedustystar-forminggalaxies(DSFGs)basedon abundancematching,whichallowsthemtopredicttheclusteringoftheanisotropies inthecosmicinfraredbackground(CIB).TheycomparedtheirpredictionstoCIB auto-spectraandCIB-convergencecross-spectrafromPlanck( PlanckCollaboration etal. 2013c a )and Herschel /SPT( Holderetal. 2013 ).TheresultwasthattheCIB auto-spectrawerewellreproduced,buttheobservedcross-spectrawerelowby 40% Thisdiscrepancywasnotresolvedin B etherminetal. ( 2013 ),andisverysimilartoour ndings.Herewecommentonpossiblecausesforthiseffect: Issueswiththelensingmap :Iftheamplitudeoftheconvergencehasbeen underestimated,itcouldexplainthelowbiasvluemeasurehere.However, van Engelenetal. ( 2012 )computedtheautospectrumofthisconvergencemapsand foundthatitwaslowwithrespecttoWMAP7concordancecosmologybyabout 14% ,whichinacrossspectrumwithgalaxieswouldimplyadecrementofonly 7% .Thisisnegligiblecomparedtotheinconsistencywemeasure. Issueswith W g ( z ) :Theredshiftdistributionofgalaxiesweuseisderivedby matchingtheIRACdatatotheCOSMOSphotometricredshiftcatalogs,as describedinChapter 2 .Therealdistributionofoursourcesmightbeslightly 94

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200 400 600 800 10 2 10 1 10 0 10 1 C gg l 10 6 Galaxies X Galaxies 0 200 400 600 800 1000 l 10 2 10 1 10 0 10 1 C g l 10 7 Lensing X Galaxies Figure4-3. Top:Measuredangularpowerspectraforthegalaxyautocorrelation.The solidlineistheexpectedlargescalesignalusingthegalaxybiasdetermined fromthegalaxy-galaxycorrelationinChapter 2 ,whichatthepivotredshiftof z =1.5 becomes b gg =2.2 0.1 .ThiscurveincludestheexpectedPoisson noiseoffset.Thereasonwhythecurveliesbelowthehighl pointsis becausewehavenotincludedthenon-negligibleone-haloterminthisplot. Bottom:Galaxy-convergencecrosscorrelation.Thesolidlinerepresentsthe bestttothepoints,nding b g =1.3 0.3 .Thedashedlineistheexpected signalassumingthegalaxybiasderivedfromthegalaxy-galaxycorrelation, showingaclearinconsistency. 95

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different,butnotenoughtoalter b gg signicantly.InChapter 2 thiswasexplored, estimatingthecosmicvarianceandusingadifferentreferencesurvey(EGS,Barro etal.2011)toderive W g .Inallcases,thedistributionremainedpeakedat z =1.5 yieldingconsistently b gg > 2 .Since b gg $ ( W g ) 2 and b g $ W g ,theratio b gg / b g islesssensitivetochangesin W g ( z ) than b gg .Awaytoobtain b gg / b g =1 by altering W g ( z ) ispossiblebymakingthisredshiftdistributionsharplypeakedat z =1.5 .However,therequiredsharpnesswouldbeunreasonable,consideringthe magnitudeofthescatterintheinfraredselectionofourgalaxies. Issueswithcosmology :Atfacevalue,ourresultscouldimplythatgalaxies donotonlytracedarkmatterinthewaytheycluster.However,thiscontradicts thecurrentparadigmincosmologythatrequiresgalaxiestoforminregionsof highmatterdensity.Thehighlevelofinconsistencywemeasure,ifattributedto cosmology,wouldrequireverylargeadjustmentsinthelatter.Thus,webelievethis scenarioisunlikely.Moreover,thestudyof Sherwinetal. ( 2012 )isverysimilarto ours,usingquasarsthathavearedshiftdistributionandbiasthatarecomparable toourgalaxies.Theyndfullagreementbetweenthebiasmeasuresfromtheauto andcrossspectra. Therefore,wedonotknowwhatmaybecausingthisdiscrepancy.Itmightrepresent ararestatisticaluctuation,butthesimilarndingsin B etherminetal. ( 2013 )warrant amorein-depthlookintothe b gg # b g testasatooltocalibrateourmodels.Also, datafromtheDarkEnergySurveywillsoonrendermuchmoreprecisemeasurments ofthebiasandotherpropertiesofthegalaxypopulation.Inaddition,lensingmaps willcontinuetoimprovewithdatafromtheSPTpolinstrument( McMahonetal. 2009 ). Theseadvancesmightbeabletodeterminewhetherourresultshaveaphysicalorigin orarecausedbyaspuriousandunaccountedeffect. 96

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CHAPTER5 DYNAMICALMASSOF Z =1 GALAXIES 5.1Background Theevolutionofthestructuralpropertiesofmassiveearly-typegalaxies(ETGs) iscurrentlyatopicofdebatewithmanyopenquestions.Inrecentyears,studieshave shownthatgalaxieswith M & 10 11 M aresmallerbyafactoroffourat z & 1.5 ( Daddi etal. 2005 ; Trujilloetal. 2006 ; Longhettietal. 2007 ; Buitragoetal. 2008 ; Damjanov etal. 2009 )andafactoroftwoat z 1 ( Trujilloetal. 2007 2011 )thanthenearby populationofthesamemass.Forthemostcompactandmassivegalaxiesinparticular, Trujilloetal. ( 2009 )foundthatonly 0.03%ofSDSSgalaxieswith M & 8 ) 10 10 M at z + 0.2 areETGswith R e + 1.5 kpc.Thiscontrastswiththemuchhigherfraction 15% ofsuchgalaxiesfoundat z 1byTrujilloetal.(2007,hereafterT07).Mostinterestingly, theluminosity-weightedagesofthelocalcompactETGsfrom Trujilloetal. ( 2009 ) areverylow( 2Gyr),whicharguesagainstthissamplebeingthecounterpartofa passivelyevolvedpopulationofhighredshiftgalaxies.Thus,thissuggeststhatmost highredshiftcompactandmassivegalaxiesmusthaveundergoneasystematicgrowth insize. Amongthedifferentmechanismsproposedtoexplainthisexpansion,theonethat reproducesbesttheobservedevolutionoftheaveragemass-sizerelation( Trujilloetal. 2011 )istheminormergerscenario( Naabetal. 2009 ; Hopkinsetal. 2010 ).Withthis mechanism,galaxiesgrowinside-outbyaccretionofsmallersatellitesthatbuildup anextendedenvelope( vanDokkumetal. 2010 ; Hopkinsetal. 2009 ).Thismodel predictsasubstantialgrowthinmassandeffectiveradius,withamilddecreaseinstellar velocitydispersion # .Thus,measurementsof # atdifferentredshiftsareadirectway toconstrainthisevolutionaryscenario.Inaddition,theyallowforanindependentmass estimatethroughthevirialtheorem. 97

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Theminormergerscenarioisconsistentwiththecommonndingsof # 250 kms 1 forgeneralETGsamplesat z =1 # 2 ( Gebhardtetal. 2003 ; Treuetal. 2005 ; vanderWeletal. 2005 ; diSeregoAlighierietal. 2005 ; Cenarro&Trujillo 2009 ; Cappellarietal. 2009 ; Fern andezLorenzoetal. 2011 ).Conversely,itisgreatly challengedbythedramaticevolutionrequiredforthesuper-densegalaxies( R e < 1 kpc, M & 10 11 M ),whichareexpectedtohavemuchlarger # .VanDokkumetal.(2009) measured # 500 100kms 1 fromasuper-denseETGat z =2.2 ,yieldingadynamical massthatisinagreementwiththephotometricmassestimate.However,more # measurementsoftheseextremegalaxiesareneededtosupportorweakentheneed ofastrongerevolutionaryscenario.InthisChapterwepresent # 'soffourmassiveand compactETGsat z 1 ,twoofwhicharesuper-dense.Wecomparedynamical/stellar massesandestimatetheevolutionarypathsoftheseobjects.ThroughoutthisChapter weadoptastandardcosmologyof m =0.3 ! =0.7 and H 0 =70kms 1 Mpc 1 5.2Dataset 5.2.1SampleSelection OurtargetswereselectedfromthesamplebyT07,whichcontainsphotometrically derivedparametersfor831galaxieswith M & 10 11 M upto z 2 intheEGSeld fromthePOWIR/DEEP-2survey( Bundyetal. 2006 ).Thoseauthorsmeasuredeffective radiiwiththeGALFITcode( Pengetal. 2002 )andderivedtotalstellarmassesby ttingspectralenergydistributionsfrom Bruzual&Charlot ( 2003 )withChabrierIMFon BRIJK photometry.WeselectedfourmassiveandcompactETGsat z 1 withS ersic index n > 4 .Threeofthemhaveextremedensitiesinthesensethattheyarethemost compactatagivenmass.Thecatalogparametersofourselectionaredisplayedinthe rstvecolumnsofTable 5-1 5.2.2ObservationsandReduction Weobserved2galaxiesatatimeusingopticallong-slitspectroscopywiththe OSIRIScamera-spectrograph( Cepa 1998 )atthe10.4mGranTelescopioCanarias. 98

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Weuseda0.8 ## slitandR1000grisminthespectralrangeof 5000 # 10000 Aundera seeingof 0.7 ## .Thetotalintegrationtimepertargetwas12800s.Theinstrumental resolutionwasdeterminedbyGaussianttingofthestrongskyemissionlinesyielding # inst =120kms 1 .However,inoneoftheobservingnightsthespectrographsuffered adefocusthatdecreasedtheresolution,affectinghalfoftheexposuresof899and 453.Fortheseframeswefound # inst =210kms 1 ,andwereducedthisdatasubset independently.ThestandardstarL970-30wasalsoobservedtouxcalibratethe spectra.ThedatareductionwasperformedwiththeIRAFlongslitpackageandinvolved thestandardstepsofbiaslevelcorrection,spectralat-elding,cosmicrayremoval, skysubtraction,frameco-addingandapertureextraction.Thetelluricabsorptionband around7600 Awaspresentinallofourspectra,affectingtheCaH+Klinesintargets 899and321.Tocorrectfortelluricabsorptioninthesetwogalaxies,weusedaF-type starobservedinthesameslitas899.Thisstar'sspectrumhasasignal-to-noiseratio (S / N)=280 andasmoothcontinuumwithnospectralfeatureswithinthetelluric band.Afteritscontinuumremovalandnormalizationweobtainedtheatmospheric transmissionproleinthespectralwindowanddividedthegalaxyspectrawithaffected CaH+Klinesbythistransmissionprole.Wedidnotapplythetelluricbandcorrection ontheothertargetsbecausethemainspectrallinesusedtomeasure # werenot affectedinthem.Notethatthiscorrectionincreasesthenoiseinthespectralwindow whereitisapplied.Thus,inthesecasesthecorrectiondidnotcontributetowardsmore precise # measurementsandforthisreasonwesimplyexcludedtheaffectedrangein wavelengthfromtheanalysis.Thenalrest-framespectrahave (S / N)=22 # 29 per AandareshowninFig. 5-1 .NotethestrongabsorptionintheCaH+Klinesandthe BalmerseriesfromH toH 12 .[OII] 5 3727 Adoubletemissionwasfoundintwoobjects. In790,thisemissionshowsadoublepeakwithaseparationof4.1 Aascomparedto 2.7 Aofthedoublet. 99

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5.2.3VelocityDispersions Tomeasure # 'sweusedthe MOVEL codewithintheR E D uc m Edistribution( Cardiel 1999 ).Thisprogramcomputesanoptimaltemplatebycreatingalinearcombination ofmodelstellartemplatespectrathatyieldsthebestttothegalaxyspectrum.The linearcombinationcoefcientsofthetemplatesweightheircontributiontothetotal stellarpopulationandprovideanestimateoftheageandmetallicitydistributionofthe galaxy.Oursinglestellarpopulation(SSP)templateswereobtainedfromtheMILES library 1 ( S anchez-Bl azquezetal. 2006 ; Vazdekisetal. 2010 )assumingChabrierIMF andbroadenedtomatchtheresolutionofourgalaxyspectra.Thesetemplateswere selectedwithmetallicities[Fe/H] = { 0.0,0.2 } andarangeofages 0.5 # 5 Gyr.Weruled outlowermetallicitiessincetheyareveryraregiventhemassesofourgalaxies( Zahid etal. 2011 ). Wettedeachspectrumusingarest-framerangeofapproximately 3700 # 4400 A sincethenoisegeneratedbythesubtractionofskyOHlineswashedoutmostfeatures intheredderpartoftheobservedwavelengthrange.Whereapplicable,weexcluded the[OII]emissionanduncorrectedtelluricabsorptionfromthe MOVEL ts.Inorderto accountforthedominantphotonnoiseinthenal # results,weperformedMonteCarlo computationswith200bootstrappedspectra.Theoutputproducedadistributionof values,whosemeanandstandarddeviationgivethevalueanderrorofthe # .Forthe objects899and453wecalculated # astheweightedmeanofthevaluesderivedfrom thetwodifferentinstrumentalresolutions,whichfavorsthehigherresolutionvaluedue toitssmallererror.ThetstothespectraaredisplayedinFig. 5-1 .Following Jorgensen etal. ( 1995 ),weapplyanaperturecorrectiontoadiameterof1.19 h 1 kpc,which increasesourvaluesaround6%.Thenal # 'saregiveninTable 5-1 .Wenotethat thereexistspubliclyavailableDEEP2spectraforallofourgalaxies( Davisetal. 2007 ). 1 miles.iac.es 100

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Thetarget453hasalreadybeenpartofthephotometricandspectroscopicstudyby Fern andezLorenzoetal. ( 2011 ),whond # =173 16kms 1 fromaspectrumof (S / N)=11 per A.Thisisconsistentwithourmeasurement.TherestoftheDEEP2 spectraofgalaxiesincommonwithoursamplehave(S/N) + 9 per Aandcannotbeused forreliable # measurements. 5.2.4StellarPopulations TheageandmetallicityofeachgalaxyweredeterminedfromthevaluesoftheSSP modelwiththelargestcoefcientinthelinearcombinationthatbestttedthegalaxy spectrum(seevaluesinTable 5-1 ).Thesevaluesshowrelativelylowagesintherange of 0.8 # 1.6 Gyr,withuncertaintiesestimatedtobearound0.4Gyr.Inaddition,we calculatedphotometricstellarmassesusingthedatafromtheRAINBOWdatabase 2 ( Barroetal. 2011a ),whichincludesphotometryforourgalaxiesin 30bandscovering arangefrom150nmupto70 m.Wetthesedatatostellarpopulationmodelsby Bruzual&Charlot ( 2003 )withChabrierIMF,assuminganextinctionlawfrom Calzetti etal. ( 2000 ).Weconstrainthemetallicityinthesamewaythanforthe MOVEL templates. Theresultsshowsolarmetallicityforallgalaxiesandagesintherangeof0.7-1.7Gyr. Suchoutcomeisverysimilartotheresultsfrom MOVEL Thegalaxies790and321have[OII] 5 3727 Aemission,implyingongoingor recentstarformation.Wecombinethe[OII]uxesfromourspectrawiththerest-frame k -corrected24 muxesfromRAINBOWtoderivestarformationrates(SFR)usingthe conversionfrom Kennicuttetal. ( 2009 )basedonKroupaIMF.ThetotalSFR [OII]+IR for the[OII]emissiongalaxiesis9.7and22.6 M yr 1 ,respectively. 5.3DiscussionandSummary Dynamicalmassesarecalculatedundertheassumptionofhomologyas M dyn = R e # 2 / G with =5 (seeTable 5-1 ),followingstudiesoflocalETGs( Cappellarietal. 2 rainbowx.s.ucm.es 101

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Table5-1. TherstvecolumnsarepropertiesfromT07,wheretheeffectiveradiiandstellarmasseshaveuncertaintiesof 0.06dexand0.2dex,respectively. M R referstothestellarmassderivedwithphotometryfromtheRAINBOW database.SB e istherest-frame B -bandaveragesurfacebrightnesswithintheeffectiveradius.Ageand metallicityareluminosity-weightedbythelinearcombinationofSSPmodels. IDRedshift I (mag) R e (kpc)log M ( M ) SB e (mag)log M R ( M ) # (kms 1 )log M dyn ( M ) Age(Gyr)[Fe/H] 7900.965621.890.51411.0715.8210.65 0.03156 1010.16 0.080.80.0 3210.915921.342.46211.1018.9510.67 0.04166 1210.90 0.081.00.0 8990.932521.710.46911.3415.6310.80 0.06236 1710.48 0.081.60.2 4530.905620.911.60111.6017.6010.98 0.05186 1010.80 0.071.30.2 102

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Figure5-1. Spectraofoursamplegalaxies.TheblacksolidlinesrepresenttheMOVEL ttingresults.Lightgreyregionsare[OII]emissionat3727 Aandtelluric absorptionat7600 A,whichwereexcludedfromthets(seetextin ¤ 2.2). 103

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! ! ! ! " " 0.0 1.0 10.0 10.5 11.0 11.5 !" log M "! 10 10 M # ! ! ! ! $ $ $ $ $ $ $ $ % % % % % % % % % % % % % 0.5 0.0 0.5 1.0 8.0 8.5 9.0 9.5 log R e "! kpc 0.32SB e # 1.25log $ & Trujillo07atz % 1, M phot & Oursample, M phot Oursample,estimated M phot Evolutiontoz 0 % Gebhardt03 $ VanderWel05 a b Figure5-2. a) Stellarmass-sizerelationat z 1 .Smallcirclesrepresenttheentire samplefromT07,blacksquaresareourselectionamongthem.Thestraight lineisthelocalrelationfrom Shenetal. ( 2003 ).Blackdiamondsareour galaxieswithstellarmassesderivedas 0.7 M dyn .Graydiamondsaretheir positionwhenevolvedto z =0 viaminormergergrowth. b) B-band FundamentalPlaneofETGs.Thesolidlineisthelocalrelationfrom Jorgensenetal. ( 1996 ).Oursampleisshownasblackdiamonds.Gray diamondsaretheirpositionwhenevolvedto z =0 viaminormergergrowth andpassiveluminosityfading.Samplesfrom Gebhardtetal. ( 2003 )and van derWeletal. ( 2005 )intherange z =0.8 # 1.1 areshownforcomparison. Wetransformedour I to B magnitudesappling k -correctionsbasedon V # I colorasin Gebhardtetal. ( 2003 ).Inaddition,weappliedanegativeoffsetof 0.1magonallsurfacebrightnessdatafrom Gebhardtetal. ( 2003 )to transformtheirVegatoourABmagnitudesystem. 104

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2006 ).Asnotedin ¤ 2.2,thegalaxy790showsadoublepeakinthe[OII]emission. Ifsuchfeatureisduetorotationofagaseousdisk,ityieldsaprojectedvelocityof V r sin( i ) 165kms 1 ,whichiscomparabletoitsmeasured # =156kms 1 .However,it isunlikelythatthiscausedanimportantbiasinthedynamicalmassestimates.At z =0 ithasbeenfoundthat =5 isrobustagainstrotationupto V r / # 1 ( Cappellarietal. 2006 2007 ),andthereisnoclearevidencethatthiswouldbedifferentat z =1 ( vander Wel&vanderMarel 2008 ).Therefore,weassumethatanydeviationsfromhomology duetopossiblerotationalsupportarewithintheerrorsofourdynamicalmasses. Ourgalaxieshavevaluesthataresystematicallysmallerthantheoriginally publishedstellarmassesbyanaveragefactorof 6.Thisismuchlargerthanthe combinederrorofphotometricstellarmassandradius(typicallyafactorof 3).Note thatforthegalaxy321thesemassesareconsistentwithinuncertainties.Thestellar populationmodelsweusedwiththeRAINBOWdatayieldstellarmassesintherangeof log M =10.65 # 10.98 ,onaverageafactorof 1.8largerthanthedynamicalmasses. ThemassesfromRAINBOWarederivedwiththesameIMFandstellarpopulation modelsasinT07butwithabroaderspectralrange(UV-FIR)asdescribedin ¤ 2.4. Inordertoinvestigatethelargedifferencesbetweenbothsetsofmasses,wealso derivedmassesusingonlyBRIJKbandsasinT07.Wefoundnosignicantdifference betweenourUV-FIRandBRIJKmasses,implyingthatthediscrepancywithT07 cannotbesimplyexplainedbytheuseofadifferentspectralrange.Wealsochecked thatourmassdeterminationsareweaklysensitivetothechoiceofextinctionmodel andsmallvariationsinmetallicity. Barroetal. ( 2011b )derivemassesforhundreds ofgalaxiesinT07'ssampleusingthefullRAINBOWphotometry.Theyndthat,on average,theirmassesareconsistentwiththosefromT07.Therefore,thecauseofthe inconsistencywithourphotometricmassesisstillunclear.Giventhelargeuncertainties inthesystematicsofphotometricmasses,weadoptdynamicalmassesasthecorrect 105

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totalmassestimates.Followinglocallensingstudies( Gavazzietal. 2007 ),weapplya ducialdarkmatterfraction f DM =0.3 andrecalculatethestellarmassesas 0.7 ) M dyn ComparisonstudiesbetweenstellaranddynamicalmassoflessdenseETGs thanthoseinoursamplehavebeenconductedatlow( Cappellarietal. 2006 )and highredshifts( vanderWeletal. 2006 )withtheresultofdynamicalmassesbeingon averageequalorlargerthanstellarmasses,asexpectedwhenconsideringadark mattercontribution.When vanderWeletal. ( 2006 )calculatestellarmassesusing similarmulti-bandphotometryandstellarpopulationparametersasinT07,theynd M / M dyn =0.66 (SalpeterIMF).Therefore,ourresultspointtowardsasystematic overestimationofthestellarmassesusedtoselectthegalaxiesofoursample.We cannotruleoutadeviationfromhomologythatrequires 0 =5 .However,thetheoretical expectationforobjectswithsuchhighS ersicindexis 5 ( Cappellarietal. 2006 ; Bertinetal. 2002 ),whichwouldyieldevensmallerdynamicalmasses. Fig. 5-2 ashowsthepositionofourobjectsinthestellarmass-sizerelationwiththe old/newstellarmassestimates.Fig. 5-2 bshowsourobjectsintheFundamentalPlane (FP).Atagiven R e ,theyhavesystematicallylargermassesandsmallercombination ofsurfacebrightness(SB e )and # thantherespectivelocalrelations.Moreover,our objectsfollowatrendinwhichtheyliefurtherfrombothlocalrelationsatsmallerradii. Thisresulthasalsobeenfoundin z 0.9 clustersby Jrgensenetal. ( 2006 2007 ). ComparedtotheFPderivedbytheseauthors,atagiven R e oureldsampleshows asimilarslopeandarelativeoffsetof 0.3 magtowardsbrighter SB e ,whichisin agreementwitheldgalaxieshavinggenerallyyoungerstellarpopulationsthanclusters ofthesameredshift. InordertounderstandhowourgalaxiesrelatetolowredshiftETGs,weinvestigate thesimpleevolutionaryscenariobasedonpassiveluminosityfadingofthestellar populationsandgrowthviaminormergersthathasbeenproposedforthegeneral populationofETGsat z 1 ( Gebhardtetal. 2003 ; Naabetal. 2009 ).Toderivethe 106

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amountoffadingforoursample,weuse Bruzual&Charlot ( 2003 )modelswithChabrier IMFandtheredshift,agesandmetallicitiesfromTable 5-1 .Passiveevolutionfrom z 1 to z =0 resultsinanaveragedimmingof 2maginSB e .Weassumethatthepassive luminosityevolutiondoesnotaffect R e .Minormergersareveryefcientgrowingthesize atagivenmassincrement,slightlydecreasing # .Accordingtothesimulationsby Naab etal. ( 2009 ),anETGwith M =10 11 M and R e =1.5 kpcwouldgrowafactorof1.5in massviaminormergersfrom z 1 to z =0 .Weadoptthisfactorofmassgrowthforour galaxies.Followingtheseauthors,wemodeleachmergeras # 2 f / # 2 i =(1+ 4/ ) / (1+ 4 ) and R e f / R e i =(1+ 4 ) 2 / (1+ 4/ ) ,wherethesubindexes(i)/(f)denoteinitial/nalvalues ofthehostgalaxyand / = # 2 a / # 2 i andthemassratio 4 = M a / M i takeintoaccountthe accretedsystem( M a # a ).Weassumeahistoryoffourmergerswithconstant 4 =0.1 and / =0.2 forallgalaxies(notethat 4 andthetotalmassgrowthsetthetotalnumberof mergers).Thischoiceofvaluesisconsistentwiththesimulationsby Naabetal. ( 2009 ). Thetotalchangefor R e and # becomefactorsof2and0.8,respectively.Theseresults arerobustagainstvariationsinthevaluesof 4 and / atthegiventotalmassgrowth.We assumethatthemass-to-lightratiodoesnotchangesignicantlyaftereachmerger.In addition,weassumethatasthesizeincreases,SB e decreasesby SB e $ 2.94" log R e ( Hamabe&Kormendy 1987 ). Fig. 5-2 showsthecombinedevolutionofminormergersandluminosityfadingfor ourgalaxies.MostofthemreachpositionsthatareconsistentwiththelocalFPand mass-sizerelations,consideringtheestimateduncertaintiesinourmeasurements. Therefore,thissimpleevolutionaryscenariocanplausiblydescribetheevolutioninour samplegalaxiestobecomenormalETGsat z =0 107

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CHAPTER6 CONCLUSIONS Ihavepresentedarobustmeasurementonthelinkbetweendarkmatterhalos, stellarmass-selectedgalaxiesat 1 < z < 2 andinfraredemissionfrom 0 < z < 4 star-forminggalaxies.ThishasbeendoneundertheHODframework,modelingthe galaxyoccupationofhaloswitharemarkablysimple(yetwell-motivated)setofanalytical prescriptions.Giventhehighredshiftofthesample,Ifoundthatthistypeofmodeling isabletodescribethedatawithreasonableaccuracy,whichsupportsthevalidityof predictedevolutioninquantitiessuchasthehalomassfunction,haloconcentrations anddensityproles.Additionally,incombinationwithCMBlensingdata,Iwasableto performacross-checkonthepredictedoverlapbetweengalaxyandmatterdistributions. Ifoundthatthespatialover-densityofthegalaxydistributioninthisregionofthesky isnotlinearlyproportionaltothatofthetotalmatter.However,thiscouldbeduetoan unaccountedsystematiceffect,andfurtherinvestigationisneededtodetermineifthis resultiscausedbyanunaccountedsystematiceffect. Ifoundthatgalaxiesat z =1.5 arehighlybiasedwithrespecttodarkmatter, comparedtothevaluesofsimilargalaxiesinthelocalUniverse.Thisbiasalsorises steeplywithstellarmass,implyingthatgalaxyformationbecomesincreasinglyrestricted tohigheroversensitydarkmatterpeakswhenlookingathigherredshiftoflargerstellar masses.Furthermore,Ideterminedaclearevolutioninthecharacteristichalomass wheregalaxyformationismostefcient.Thishalomassisalmost5timeslargerthan at z =0 ,andcouldbecausedbythedropincosmicdensityastheUniverseevolves, whichmakesitmoredifculttoformmassivegalaxies.Thosemassiveonesat z =0 hadmostoftheirstarsalreadyinplaceat z 1 ,whilesmallergalaxiesmaintaineda higherspecicgrowththroughoutthisredshiftrange.Ialsoinvestigatedthestellarmass buildupof z =1 galaxiesofagivensize.Bycalculatingmassesbasedonthedynamics oftheirstars,IwasabletocorrectpreviousmeasurementsbasedonSEDttingand 108

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foundthatafewminormergersareaplausiblewaytoevolve z =1 galaxiesontothe localmass-sizerelation. 109

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APPENDIXA PHOTOMETRICSIMULATIONS Ourselectioncriteriaarebasedonaperturephotometrydrawnfromthe Ashbyetal. ( 2013 )4.5 m-selectedcatalog,butweusedsourcesfainterthantheir5 # sensitivity threshold.Forthisreason,wecarriedoutanindependentanalysisoftheSSDF sourceextractioninordertoestimatethecompletenessandphotometricbiasfor thefaintestsources.Thiswasdonebyplacingarticialsourcesofknownbrightnessin representativeSSDFmosaics(i.e.,apairofcoextensive3.6and4.5 mtiles'ofsize 2 ) 1 deg 2 ;Ashbyetal.2013b),whichwereobservedtothenominalsurveydepth.Then, weperformedadetailedcomparisonbetweentheresultingphotometricmeasurements andtheinputbrightnesses,asdescribedbelow. A.1SimulationProcedures Webeganbygeneratingpointspreadfunction(PSF)imagestorepresentthe articialsources.Thisisconsistentwithhigh-resolution HST /WFC3observations showingthatthevastmajorityofgalaxieshavingmagnitudesinourrangeofinterestare pointsourcesatIRACresolution(Fig.25ofAshbyetal.2013a).Werstidentied18 pointsourcesintheSSDFsciencemosaicsandveriedbyvisualinspectionthatthey didnotcontainanyartifactsoranomalies.Thesepointsourceswerethenscaledand median-stackedattheircentroidpositions.TheresultingPSFimageswereconstructed with41by410.6 ## pixelstomatchthespatialresolutionoftheSSDFsciencemosaics. TheFWHMsoftheseimageswerefoundtobe1.69 ## and1.85 ## inthe3.6and4.5 m bands,respectively.ThesevaluesareclosetothosemeasuredforIRACinsingle exposures,i.e.,1.62 ## and1.77 ## ThePSFimageswerethenplacedinthesciencemosaic.Theywereplacedat randompositions,butataminimumdistancefromeachother.Thisminimumdistance varieslinearlywiththemagnitudeofthesimulatedsourcefrom18 ## to6 ## through [4.5]=15 # 21 .Inaddition,simulatedsourceswerenotallowedtofallwithinregions 110

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15 16 17 18 19 Input Magnitude 0 5 0 4 0 3 0 2 0 1 0 0 0 1 0 2 0 3 Photometr ic bias (mag) 5 [4.5] [3.6] FigureA-1. Photometricbiasinthe3.6and4.5 mSSDFmosaics.Thephotometric biaswasmeasuredasthedifferencebetweentheinputmagnitudeandthe outputaperture-correctedmagnitude,asderivedfromthesimulations.The solidlinesrepresentthemedianbiasanderrorbarsareonestandard deviation.Thearrowmarksthe 5 # sensitivitylevelin[4.5]uxes.Thebias trendin[4.5]indicatesanarticialbrighteningofsourcestowardsfaintinput magnitudes,mainlycausedbythecontaminationofuxfromnearbyobjects andnoisepeaks.Ontheotherhand,[3.6]extractedmagnitudesbecome slightlyfaintertowardsfaintinputmagnitudes,whichisduetopositional shiftsinthe[4.5]selectionaperturewithrespecttotherealcentroidofthe source.Theaperture-correctedmagnitudesintheSSDFcatalogare combinedwiththephotometricbiastoobtainthenalphotometry(seetext). 111

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15 16 17 18 19 20 21 [4.5] magnitude 0 0 0 2 0 4 0 6 0 8 1 0 Reco v er y fr action 50% 80% 5 Ashb y13 FigureA-2. Recoveredfractionofsimulatedsourcesasafunctionofinputmagnitudein [4.5].Thedash-dottedlineat18.19representsthe5 # levelofphotometric sensitivity.Thedashedanddottedlinesmarkthepointswheretherecovery fractionreaches80%at18.58and50%at19.39,respectively.Open trianglesrepresenttherecoveryfractionderivedin Ashbyetal. ( 2013 ).We measurehighercompletenessthanseenby Ashbyetal. ( 2013 )becausewe employamorecomplicatedsourceidenticationprocedure,whichboosts thedetectionsincomplexcases(e.g.,blends,positionalshifts)that otherwisewouldberejected. contaminatedbystarsbrighterthan K s =12 mag.Thesizeoftheexclusionregions aroundthesestarswasdeterminedfollowingthemethoddescribedinSection 2.2 Outsidethemaskedregions,1500simulatedpointsourceshaving[4.5]=15mag wereaddedtorandomlocationsofthe4.5 msciencemosaic.Thisnumberof simulatedsourcesislowenoughtoavoidcrowdingandalterationsinthephotometric background.Anequalnumberofsourceswerealsoputatidenticallocationsinthe 112

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3.6 msciencemosaic.The3.6 msourcesweresettohavecolors [3.6] # [4.5]=0.7 appropriateforthesampleselectiondescribedinSection 2.3.2 .Themodiedscience mosaicswerethenprocessedwithSExtractor.Thiswasalsodoneontheoriginal, unmodiedsciencemosaic.IdenticalSExtractorparametersettingswereusedinall instances,followingthosepresentedin Ashbyetal. ( 2013 ).Theprocesswasrepeated untilatotalof80,000[4.5]=15magsourcesweredetectedandphotometered.The simulationswerethencarriedoutinthesamemannerforinputmagnitudesintherange [4.5]=15.5 # 21 magwithstepsof0.5mag.WeusedtheresultingpairsofSExtractor catalogstodetermineourdetectioncompletenessandphotometricbias.Specically,we retrievedallcatalogedsourcesfoundwithin6 ## ofthepositionofeachsimulatedsource inboththeoriginalandmodiedmosaics.Thissearchradiuswassetempiricallyto encompassallpossibleshiftsinthemeasuredcentroidsofsourcesduetothedistortion causedbythesimulatedsource.Sourcesinthetwocatalogswerejudgedtomatch whentheydifferedbylessthan50%inuxandwereseparatedbylessthanhalfthe PSFFWHM(0.9 ## ).Thisleftanumberofnon-matchedsourcesfromtheoriginaland modiedmosaics: N orig and N mod ,respectively.Then,adetectionofthesimulated sourcewasdeterminedifoneofthefollowingcasesapplied: A: N orig =0 N mod =1 .Thisisthemosttypicalcase,whereonlyonesourceinthe modiedmosaiccouldnotbematchedtoanotherintheoriginalmosaicandwas thereforeidentiedasthesimulatedsource. B: N orig =0 N mod > 1 .Asin A ,allsourcesintheoriginalmosaicwereuniquely identiedinthemodiedmosaic.However,therewereafewsourcesinthe latterwithnocounterpart.Thishappenedbecausethesimulatedsourcewas erroneouslyrecoveredbySExtractorasmultiplesources.Weidentiedthe brightestoneofthesewiththesimulatedsource. C: N orig > 0 N mod > N orig .Thelocalphotometricinuenceofthesimulatedsource causedachangeinthepositionanduxofseveralsourcesinthemodied 113

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mosaic.Asaresult,notallsourcesintheoriginalmosaicwerefoundamatchin themodiedmosaic.Thisleftatleasttwocandidatesinthemodiedmosaicto representthesimulatedsource.Consequently,oneofthemwouldcorrespondto anunmatchedsourceintheoriginalmosaic.Tosimplifytheidenticationprocess, werestrictedtheselectionofunmatchedsourcesbasedontheinputlocationof thesimulatedsource:weconsideredtheclosestoneandtheclosesttwofromthe originalandmodiedmosaics,respectively.Ifthesourcefromtheoriginalmosaic wasbrighterthanthesimulatedone,thentheformeroughttohaveshiftedits positionlessthanthelatter.Thus,thesourceintheoriginalmosaicwasmatched tothesourceinthemodiedmosaicthatlayclosesttoit.Otherwise,thesimulated sourcemayhavenotshiftedsignicantlyduetothepresenceofthesourceinthe originalmosaic.Inthiscase,thesimulatedsourcewasidentiedwiththenearest sourceinthemodiedmosaic. D: N mod > 0 N orig & N mod .Inthissituation,eitherablendoccurredorthesimulated sourcedistortedthelocalbackgroundinsuchawaythatsomesourcesinthe originalmosaicwerenotrecoveredinthemodiedmosaic.Inthelattercase, thecandidatetorepresentthesimulatedsourcewastheonefoundclosestto it.Adetectionwasjudgedifthecandidatewastheresultofablendbetween thesimulatedsourceandoneormoreintheoriginalmosaic,providedthatthe simulatedsourcedominatedthetotalux.Thiswasconrmedwhentheuxratio betweenthecandidateandthesimulatedsourcewaslessthan2. Adirectproductofthesephotometricsimulationsistherelationbetweeninput and4 ## -aperturerecoveredmagnitudes.Forbrightsources,thisquantityshouldmatch theaperturecorrection,whichrepresentstheuxlossduetotheniteaperturesize. The4.5 m-selectedcatalogfrom Ashbyetal. ( 2013 )includesaperturecorrections of ( [3.6] [4.5] )=(0.33,0.33) ,derivedfromPSFgrowthcurves.Thevaluesreturned byoursimulationsfor [4.5]=15 magsourcesare ( [3.6] [4.5] )=(0.32,0.36) .For 114

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consistencywiththerestofourprocedures,weundothecorrectionsappliedin Ashby etal. ( 2013 )andusethevaluesderivedhere. A.2PhotometricBias DuetothebroadPSF,IRACphotometrysufferedfromanon-negligiblelevelof sourceconfusion.Thiscausedtheapertureuxtobecontaminatedbyneighboring sourcesandphotometricbackgroundnoise.Ingeneral,theseeffectsweremore signicantforfaintersources.Thusitwasnecessarytomeasuretheaveragephotometric biasasafunctionofmagnitude,anduseittocorrectthefullSSDFsourcecatalog.For thispurposeweemployedtheresultsfromthesimulationsdescribedinAppendix A.1 Wecalculatedthephotometricbiasasthemediandifferencebetweentheinput magnitudeofthesimulatedsourcesandtherecoveredaperture-correctedmagnitude, forthosesourcesthatweredetected.Thephotometricbiasesfor[4.5]and[3.6](sources wereselectedin4.5 m)areshowninFigure A-1 .Thisbiasbecameprogressivelylarger in[4.5]towardfaintermagnitudes,inthesensethatthosesourceshadgreaterexcessof uxduetocontamination.Animportantcontributiontothecontaminationinfaintsources camefromthebackgroundnoiseuctuations.Sourcesfallingontopofnoisepeaks becamebrighter,whilethoseoverlappingwithnoisetroughscouldavoiddetection. Therefore,theneteffectwasanoverestimationofthe[4.5]uxinpointsources,which becameincreasinglyimportanttowardthefaintend. Inthecaseofthe3.6 mphotometry,thephotometricbiasfollowedtheopposite trendthanseenat4.5 m.Thiscanbeunderstoodasbeingdrivenbythe4.5 m selection.AsmentionedinAppendix A.1 ,afaintsourcewaslikelytobemeasured inthe4.5 mmosaicatashiftedlocationfromitstruecenter,whichwascausedby theadditionofanearbybackgrounduxspikeinthe4.5 mmosaicandboostingthe extracteduxvalue.However,withinthesameapertureinthe3.6 mmosaic,that uxspikewasnotpresent.Wehaveveriedtheseeffectsbyvisualinspectionofthe modiedmosaicsinthesimulations.Onaverage,the3.6 mmeasurementdidnotadd 115

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extrauxfromthebackgroundandstilllostinputuxduetotheapertureshift.This inducedaphotometricbiasthatslightlyunderestimatedthe3.6 mapertureuxes. ThedetectionalgorithminAppendix A.1 alreadyrequiredsomephotometricbias correctiontocomparethemeasuredandinputuxesinitems C and D .Therefore,we ranarstpassofthesimulationtocomputeapre-correction,whichwasthenused inthesecondpasstoobtainthenalresults.Intherstpassweonlyconsideredthe photometryofrecoveredsourcesvia A and B ,whosedetectionwasindependenton thephotometryitself.Inthesecondpass,weranthesimulationusingthefulldetection algorithm,whereweusedthepre-correctionstoperformcomparisonwithinputuxes. Inthisalgorithm,wedidnotuse4 ## apertures.Instead,3 ## and5 ## corrected[4.5]uxes wereusedfor C and D ,respectively.Thepre-correctionswerecomputedinthese apertures.Wechose3 ## becauseitwastheaperturewiththelowestphotometricscatter, and5 ## duetothelargersizesthatsourceblendsgenerallyhad. A.3Completeness WiththeresultsfromthemocksourcesimulationsdescribedinAppendix A.1 wecancomputethedetectionfractionasafunctionofinput4.5 mmagnitude.This photometriccompletenessisshowninFigure A.1 ,reaching80%at18.58and50%at 19.39.Weobtainsignicantlyhighervaluesthan Ashbyetal. ( 2013 ),whichisduetothe differentproceduresusedinthedetectionalgorithm.Theprocedureusedinthiswork considersalargernumberofcaseswhereasourcemaybedetected.Thisincludesthe positionalshifts > 1 ## inthemeasuredphotometryandtheuxvariations > 0.5 magdue tosourceconfusion(seeitems B D inAppendix A.1 ). Thescatterintheextracted4.5 muxesallowsustodeterminethephotometric sensitivity.The5 # limitis[4.5]=18.19,verysimilartothelevelof18.2foundin Ashby etal. ( 2013 ). Wehavealsotestedtheeffectofvariablesurveydepthonthecompleteness. Thenominaldepthofthesurveyconsistsof N exp =4 exposuresof30secondseach, 116

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butthetilingschemeproducessomecoverageinhomogeneityinscalesgoingfrom arcminutesdowntoafewarcseconds.Thismeansthatevenasinglesourcecanoccupy aregionwithdifferentdepths.Inthesecases,wecomputeaneffectivecoveragefor agivensimulatedsource,whichconsistsofthemeancoveragewithintheindividual segmentationregion. Wehavefoundthatsourceconfusiondominatesthecompletenessforalldepths N exp & 4 ,saturatingtothesamerelationasinFigure A.1 .Sourceswith effective N exp < 4 showslightlylowercompletenessvaluesatfaintmagnitudes,buttheycomprisejust 3 %ofallsources.Thus,theireffectontheoverallcompletenessisnegligibleandwe canrelyonanexcellentuniformityacrosstheentiresurveyarea. 117

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APPENDIXB COSMOSVSEGSASREFERENCESAMPLES AlltheprocedurespresentedsofarhaveusedCOSMOSdataasthereference sample,butwehavealsoperformedthesamecalculationswithEGS.Thecomparison betweenthesesetsofresultsprovidesasenseofthesystematicuncertaintyassociated withthechoiceofthereferencesurvey.Theredshiftdistributions,numberdensitiesand stellarmassespresentsomedifferences,butweshowbelowthatthesevariationsdonot qualitativelyalterourresults. First,wepresentabriefdescriptionofsomeoftherelevantaspectsoftheEGSand COSMOSdatasets: EGS : Barroetal. ( 2011a )selectsourcesinIRACwith4 ## aperturesandreachS/N =5 # at 21 magnitude.Thesurveycovers0.48deg 2 andphotometricredshifts areprovidedwithanaccuracyof & z / (1+ z )=0.034 .TheIRACphotometryinEGS isalmost3magnitudesdeeperthanSSDF,reachingahighersourcecompletness throughouttherangeofmagnitudesconsideredinthiswork( [4.5] < 18.6 ). Likewise,thehigherdepthinEGSmakesitrobustagainstthephotometricbias thataffectstheSSDF(seeAppendix A.2 ). Barroetal. ( 2011a )applyaperture correctionsof ( [3.6] [4.5] )=(0.32,0.36) derivedfromPSFgrowthcurves.These valuesareexactlythesameasourphotometriccorrectionsinthebrightlimit(see Figure A-1 ). COSMOS : Muzzinetal. ( 2013a )explainthatimagesfromoptical+NIRbands arePSF-matchedandsourceselectionisdoneinthe K s bandwith2.1 ## color apertures.The K s bandimagesarethenusedashighresolutiontemplatesin attingproceduretodeblendconfusedIRACsources.The4.5 mphotometry reachesS/N=5 # at 20 ,whichis2magnitudesdeeperthantheSSDFand thereforecompletenessinCOSMOSisnotaconcern.Thesurveyregioncovers 1.62deg 2 andthephotometricredshiftsareaccurateto & z / (1+ z )=0.013 .The aperturecorrectionsareverydifferentfromtheschemesusedinEGSandbyusin theSSDF.FromPSFgrowthcurves,theyderivethecorrectiontothe K s AUTOux. TheratiobetweenthiscorrectedAUTOuxandthe2.1 ## uxin K s istheusedas theaperturecorrectionforallotherbands. InCOSMOS,theIRACaperture-uxcorrectionsfactorsare 50%loweron averagethaninEGS,takingintoaccountthedifferentaperturesizes.Inotherwords, forthesamepopulationofgalaxies,EGSmeasuresahigherIRACapparentuxthan 118

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COSMOS.Surprisingly,thishasaverylittleimpactinthestellarmassesforagiven luminosity,wherethedifferenceisjustanoffsetof 0.07 dexbetweenthesecatalogs. InFigure B-1 wecomparetheHOD-derivedquantities b e # g M 1 / M min and f sat for EGSandCOSMOS.Thisguredemonstratestheagreementbetweenthesedatasets. NotethattheredshiftdistributionsusedtomodeltheACFsaregeneratedfromthese referencecatalogs,asdescribedinSection 2.3.1 .RegardingtheSHMR,thepeakusing COSMOSwasfoundat log M peak =12.43 0.08 ,whereasitis 12.35 0.10 forEGS. Thesemeasurementsaremutuallyconsistentgiventheiruncertainties.Inaddition,the slopesoftherelationarealsopracticallythesame. 119

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FigureB-1. ResultsfromtheHODtstotheSSDFdatausingCOSMOSandEGSas referencecatalogs.Eachpointdenotesasampledenedbyalimiting apparentmagnitudethreshold.Inthetoppanel,theshadedregions representthe 1 # intervalofdirectlarge-scalebiasts.Theseare consistentwiththeHODbias.Thereisanexcellentagreementbetween bothdatasetsinallpanels. 120

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APPENDIXC INTEGRALCONSTRAINT Theestimatorsoftheangularcorrelationfunction(suchastheoneinEquation 215 )sufferfromawell-knownsystematicsupressionduetothenitesizeofthesurvey, calledthe integralconstraint ( Peebles 1980 ).Byconstruction,theestimatorrequiresthe probabilitytobenormalizedoverthesurveyarea.Thismeansthat: & survey ( + ) d !=0. (C1) However,thetrue ( + ) isnormalizedwiththeentiresky,sothat 4 sky ( + ) d !=0. (C2) Equation( C1 )showsthat ( + ) willbedifferentfrom ( + ) wheneverthesurveyis afractionofthesky.Inordertocalculatethecorrection ( + ) % ( + ) fortheSSDF ACFs,werunsimulationswithmockrealizationsofthegalaxyeldinthesurvey region.Thesearegeneratedwithsomeknown ( + ) ,whichisthencomparedtothe measuredestimator ( + ) .Weadoptthepowerspectrumofdarkmatter P ( k z =0) andaredshiftselectionfunctionequaltothatofourmaingalaxysample, % mock ( z )= % 12 cut ( z ) / % 12 cut ( z # ) dz # (seeFigure 2-5 ).Following Tegmarketal. ( 2002 ),theangular powerspectrumiscomputedas C mock l = 2 2 $ & 0 dkk 2 P ( k ) 5 6 $ & 0 dz % mock ( z ) G ( z ) j l ( ( z ) k ) 7 8 2 (C3) where G ( z ) isthegrowthfactor, ( z ) theradialcomovingdistanceand j l isthespherical besselfunction.WeusetheroutineSYNFASTintheHEALPix 1 distribution( G orski etal. 2005 )toproduce1000differentskyrealizationsdrawnfromthisangularpower 1 http://healpix.sf.net 121

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spectrum.TheseoverdensitymapsarecroppedtotheSSDFmaskandrenormalized withinthatregion.ApixelizedversionoftheestimatorinEquation 215 (seeEquation 18inScrantonetal.2002)isthenemployedtocalculatetheangularcorrelation function.Thetheoreticalexpressionofthisstatistictakestheformof mock ( + )= % l (2 l +1) 4 2 C mock l P l (cos + ), (C4) where P l aretheLegendrepolynomials.Figure C-1 showsthistheoreticalexpectation alongwiththemeasuredstatisticfromthemocksimulations.Wedenethedifference betweenbothcurvesas ( + ) .Ingeneral,thisquantityshouldscalewiththeoverall biasoftheACFbeingcorrected.Therefore,wecorrectthegalaxyACFsas gal ( + )= gal ( + )+ 0 ( + ) .Here, 0 isarelativebiasfactordeterminedbythequotient 0 = gal / mock at + =0.5 ,where 0 .Forourlargestgalaxysample,thelarge-scalecorrectionis 10 4 .Inallsamples,thecorrectionisconsiderablysmallerthantheerrorsatallscales anditdoesnotplayasignicantroleintheresultspresentedinthisstudy. 122

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10 0 10 1 (deg rees) 0 0001 0 0000 0 0001 0 0002 0 0003 ( ) mo c k (theoretical) mo c k (measured) FigureC-1. Red :Theoreticalangularcorrelationfunctionofdarkmatterfollowinga redshiftdistributionasinFigure 2-5 Blue :Measuredangularcorrelation functionofmockgalaxyeldswithintheSSDFsurveyregion.Theirparent distributionfollowsthesamestatisticsasthetheoreticalcurve.Errorbars representonestandarddeviation.Thesuppressioninthemeasuredcurveis duetothenormalizationofthemeangalaxydensitywithinasurveyregion thatisafractionofthetotalsky. 123

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APPENDIXD THEHALOMODEL Weusethehalomassfunctionfrom Tinkeretal. ( 2010 ),whichconsidersspherically collapsedhaloeswithanaveragedensity200timesgreaterthanthecriticaldensityof theUniverse.Ittakestheform dn ( M z ) dM = 3 m M f ( 1 ) d 1 (D1) where 3 m isthecomovingaveragematterdensityoftheUniverse.Thefunction f ( 1 ) is empiricallydeterminedbysimulationsin Tinkeretal. ( 2010 )andparametrizedwiththe variable 1 ( M z )= & c ( z ) # ( M z ) (D2) Here, & c isthecriticaldensityforhalocollapse( Press&Schechter 1974 )forwhichwe adopttheredshiftevolutionfrom Weinberg&Kamionkowski ( 2003 ) & c ( z )= 3 20 (12 2 ) 2 / 3 (1+0.131log! m ( z )), (D3) withtheuniversalfractionofmatterevolvingas m ( z )= 1+ ! m 0 (1+ z ) 3 / 1 (D4) Thermsofthematterdensityeldinsidespheresof R =(3 M / 4 2 3 m ) 1 / 3 is # 2 ( M z )= G 2 ( z ) $ & 0 dk k 2 P lin ( k ) 2 2 2 W 2 ( kR ) (D5) where P lin isthelinearmatterpowerspectrumtoday, W ( x )=(3 / x 3 )(sin x # x cos x ) and thegrowthfactoris( Linder 2005 ; Weinbergetal. 2013 ) G ( z )=exp 5 6 # z & 0 dz # 1+ z # m ( z # ) 0.55 7 8 (D6) 124

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Withthemassfunctionwecanwritethepredictedtotalnumberdensityofgalaxiesas n g = M high & M low dM dn dM ( M ) N ( M ) (D7) wheretheintegrationlimitsaresethereafterby M low =10 8 M and M high =10 16 M .The NFWhalodensityproleis 3 h ( M r )= 3 s ( r / r s )(1+ r / r s ) 2 (D8) Here, r s = r 200 / c ,where r 200 = [ 3 M / (4 2 200 3 m ) ] 1 / 3 andtheconcentrationparameteris givenby Duffyetal. ( 2008 ) c ( M z )= A ( M / M pivot ) B (1+ z ) C (D9) with A =6.71 B = # .091 C = # 0.44 and M pivot =2.86 ) 10 12 M .Wehavealsotried otherconcentrationmodelsfromtheliterature( Bullocketal. 2001 ; Gaoetal. 2008 )and exploredvariationsinthenormalization.WendtheACFstoberelativelyinsensitiveto thesechangeswithintheangularscalesprobedbyourdata.Thecentraldensity 3 s can bedeterminedthrough M = r 200 & 0 dr 4 2 r 2 3 h ( M r ) (D10) sothat 3 s = 200 3 m c 3 3 [ ln(1+ c ) # c / (1+ c ) ] (D11) Forthelarge-scalehalobias,weadopttheprescriptionfrom Shethetal. ( 2001 ) 125

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b h ( M z )= b h ( 1 )=1+ 1 & c ( a ( a ( a 1 2 )+ ( ab ( a 1 2 ) 1 c # ( a 1 2 ) c ( a 1 2 ) c + b (1 # c )(1 # c / 2) / (D12) withtheupdatedparametersfrom Tinkeretal. ( 2005 ) a =0.707 b =0.35 c =0.8 Underthehalodenitionweuse(spherical-overdensity,Tinkeretal.2008),haloesare allowedtooverlapaslongasthecenterofonehaloisnotcontainedwithintheradiusof anotherhalo.Thefullscale-dependentbiasisgivenby Tinkeretal. ( 2012 ) b h ( M z r )= b h ( M z ) [1+1.17 m ( R % z )] 1.49 [1+0.69 m ( R % z )] 2.09 (D13) with R % = 9 : : : : ; : : : : < r if r > =2 R halo 2 R halo if r < 2 R halo (D14) whichsetsaconstantbiasintheregimewherehaloesoverlap.Thenon-linearmatter powerspectrumisobtainedwiththeCAMBpackage( Lewisetal. 2000 ).Ittransformsto thecorrelationfunctionas ( r )= 1 2 2 2 $ & 0 dkk 2 P ( k ) sin kr kr (D15) Sincethevirializedregimeofsatelliteswithinhaloeswillbedifferentfromthelarge-scale interactionbetweencentralgalaxies,itisconvenienttoexpressthespatialcorrelation functionasasumoftwoterms: g ( r )=1+ 1 h g ( r )+ 2 h g ( r ). (D16) Theone-halotermishighlynon-linearanddominatesatscalessmallerthanthe averagehalosize(i.e.,virialradius),whilethe2-halotermbecomesmoreimportantat large-scales.Furthermore,theone-halotermcanbeseparatedintocentral-satelliteand 126

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satellite-satelliteparts.Intheformer,thecorrelationfollowstheformofaNFWdensity weightedbysphericalvolume,sincebyconstructionsatellitesaredistributedaccording tothatprolefromthecentralgalaxy.Inthelatter,thesatellite-satellitecorrelation followstheformofaNFWprole(stillrepresentingthedistributionofsatellitesfrom thecentralgalaxy)convolvedwithitself.Inthecaseofthe2-haloterm,thecorrelation tracestheconvolutionbetween m anddensityprolesofdifferenthalos.Giventhe manyconvolutions,itisbettertoworkinFourierspace,whereallthesebecomesimple products.Thus,Equation( D16 )canberewrittenas P g ( k )= 2 P cs g ( k )+ P ss g ( k ) 3 1 h + P 2 h g ( k ). (D17) Theexplicitformofthe1-halotermsis P cs g ( k z )= 2 n 2 g M high & M low dMN s ( M ) N c ( M ) dn dM ( M z ) u ( k M z ), (D18) P ss g ( k z )= 1 n 2 g M high & M low dMN s ( M ) N c ( M ) dn dM ( M z ) u 2 ( k M z ), (D19) where u istheFouriertransformofaNFWprole( Cooray&Sheth 2002 ).Forthe 2-haloterm,wemustnotconsideroverlappinghaloesifoneoftheirradiicontainsthe centeroftheother.Thisisdonebyadoptinghalo-exclusion( Zheng 2004 ),whereweset theminimumseparationallowedfor2haloesto d =max( R halo1 R halo2 ) ( Leauthaudetal. 2011 ).Thisimpliesthatmeasuringhalocorrelationsatdistancessmallerthat r ,wecan integrateallpossiblepairswheretheindividualradiiareboundtoanupperlimit R lim = r The2-halotermthusreads 127

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P 2 h g ( k r z )= P m ( k z ) 1 n # g ( r ) 2 M lim ( r ) & M low dMN ( M ) dn dM ( M z ) b h ( M r z ) u 2 ( k M z ) / 2 (D20) wherethescale-dependenthalobiasisintroducedand M lim ( r )= M ( r = r 200 ) enforces halo-exclusion.Thisintegrationlimitrestrictstheaveragedensityofthegalaxies considered( Tinkeretal. 2005 ): n # g ( r )= M lim ( r ) & M low dM dn dM ( M ) N ( M ), (D21) comparedtothetotal n g inEquation( D7 ).AfterFouriertransforming P 2 h g into 2 h g # ,the probabilityfunctionneedstobesupressedtoaccountforthemissinggalaxiesin n # g as 1+ 2 h g ( r z )= n # g ( r ) n g 2 [1+ 2 h g # ( r z )] (D22) Addingtheonehalotermsfromeqs.( D18 D19 )completestheHODdescriptionof ourmodel. 128

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APPENDIXE LOWREDSHIFTBUMP Lowz galaxieshaveanimportantcontributiontotheredshiftdistributioninour brightestgalaxysample, [4.5] < 16.2 ,asshowninFigure 2-5 .InthisSection,weuse theopticalSuperCosmossurveydata( Hamblyetal. 2001 )tomatchandremovethese sourcesfromtheSSDFcatalogandevaluatehowthischangestheHODresultsfrom Section5.Thisisintendedtoserveasaconsistencycheckforthemethodsusedsofar tomodelthelow-redshiftgalaxyclustering. SuperCosmos(hereafterSC)isafullskysurveyproducedfromdigitized photographicplatesinthe B R and I bands,withatypicaldepthof R ( AB ) 21 mag.WeretrievedfromtheSuperCosmosScienceArchive 1 allsourcesintheSSDF footprintwithdetectionin R andatleastoneotherband.Wechose R asthemain opticalbandbecause,incombinationwithourinfraredcuts,itisparticularlyeffectivein selecting z < 1 galaxiesat R (AB) 22.5 ( Papovich 2008 ).WeremovedfromtheSSDF catalogallthesourcesintheSCsamplethatmatchedwithinasearchradiusof 1 ## .The SSDFclusteringcomputationandmodelingfollowedthesameproceduresdescribed throughSections3-6,exceptforamodicationofEquation 22 .Thisequationdescribes theSSDFredshiftdistributionasasumofcontributionsfromtheindividualgalaxiesin thecontrolsample,givenaparticular[3.6]and[4.5]selection.Themodicationconsists ofincludingaweightfactortoeachindividualcontributionbasedonthegalaxy's R -band magnitude.Thisweight, W ( R ) 1 [ 0,1 ] ,shouldrepresenttheprobabilityofagalaxyin theSSDFtobeundetectedinSC.Wedenedthisprobabilityas W ( R )=1 # U ( R ) where U ( R ) isthe R -bandcompletenessinSC.Weestimatedthecompletenessdirectly fromthedistributionof R magnitudesfromtheSCcatalog,showninthelowerpanelof 1 surveys.roe.ac.uk/ssa/ 129

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Figure E-1 .Thisdistribution, D ( R ) ,wasassumedtoobeyapower-law(Schechter1976) thatissupressedatthefaintendbythecompletenessfunction: D ( R )= d 0 R d 1 U ( R ). (E1) Here, d 0 and d 1 arethepower-lawcoefcientsand U ( R )= 1 2 ( 1 # Erf ( s ( R # R 0 ))) (E2) Erf( y )= 2 ( 2 & y 0 e t 2 dt (E3) where R 0 and s areparametersthatcontroltheshapeoftheerrorfunction.Wet D ( R ; d 0 d 1 R 0 s ) totheSCdata.ThisisshowninFigure E-1 ,wheretheupperpanel representsthecorresponding U ( R ) component. Theapplicationof W ( R ) tothegalaxiesinthecontrolsampleproducesastrong suppresionofthelow-redshiftcontribution.ThiscanbeseeninFigure E-2 forthe [4.5] < 16.2 subsample.ComparedtotheducialdistributionderivedinSection 2.3.2 ,theremovalofSCsourceseliminatesalmostcompletelythebumpat z 0.3 Thecomovingnumberdensityat z =1.5 (Equation 26 )increasesonly 7% .The HODtsuffersadecreasein M min of 0.02 dexandanincreasein M 1 of 0.05 dex.For subsampleswiththresholdsfainterthan [4.5]=16.2 ,thesevariationsbecomeeven smaller.Overall,theimpactofremovingSCdataontheresultspresentedinthispaper isnegligible.Thisprovidessolidsupporttothemodelingofthelow-redshiftgalaxy clusteringdescribedinSections3-5. WehavenotusedtheanalysisfromthisSectiontoderivethemainresultsofthe paperbecausetherearepotentialsystematiceffectsintheSCcatalogthatwehavenot thoroughlyinspected.Forinstance,theSCphotometrysuffersfromdifferentialcoverage depthacrosstheeld,whichcouldcauseanarticialcontributiontotheclustering. Inaddition,wedonothaveacompletestatisticaldescriptionofthelargephotometric 130

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19 0 19 5 20 0 20 5 21 0 21 5 22 0 0 2 0 4 0 6 0 8 1 0 Completeness 19 0 19 5 20 0 20 5 21 0 21 5 22 0 R mag (AB) 0 0 0 5 1 0 1 5 2 0 2 5 3 0 Coun ts (10 3 mag 1 deg 2 ) Fit SuperCosmos FigureE-1. Distributionof R -bandmagnitudesfromSuperCosmossourcesinthe SSDFregion(points).Wetthedatawithafunctionconsistingofapower lawtimesanerrorfunction(dashedline).Thebest-terrorfunctionis displayedinthetoppanelandrepresentsthe R -bandphotometric completenessoftheSuperCosmossample. R -banderrorspresentinSC.Animprovedtreatmentinthisanalysiswouldentailthe useofsucherrorstodeconvolvetheSCdistributionof R magnitudes,inordertobe consistentwiththehighphotometricqualityofthecontrolsample. 131

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0 0 0 5 1 0 1 5 2 0 2 5 Redshift 0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 ( z ) (arbitrary scaling ) 0.6 < [3.6] [4.5] < 0.8 [4.5] < 16.2 Fiducial SC remo v ed FigureE-2. Redshiftdistributionofourbrightestgalaxysampleobtainedfromthe COSMOS-basedcontrolsample.Thesolidlinerepresentstheducial distributionasderivedinSection 2.3 ,whichincludessourcesselectedonly withIRACdata.Thedashedlineshowsthesimulatedeffectofremovingall sourceswith R -banddetectionsinSuperCosmos.Thisopticalselectionis veryeffectiveatsuppressingthelow-redshiftbump. 132

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APPENDIXF NOPRIORONNUMBERDENSITY InthettingprocedurewehavexedallbasicHODparametersexcept M # 1 ,and usedgalaxynumberdensitiesobtainedthroughacombinationoftheSSDFobserved numbercountsandthecontrolsample.Here,wecheckwhathappensifweleaveall thoseparameterstovaryfreelyanddiscardanypriorinformationonthenumberdensity. Thus, n g becomesaderivedquantitytroughEquation D7 .Westillneedtousethe normalizedredshiftdistributionsfromthecontrolsample,however.Forthesakeofclarity inthissection,wewillcalltheducialtofthispaper modelA (1-parametert, n g xed), andtheunconstrainedone modelB (5-parametert, n g asderivedquantity). Thegoodnessoft, 2 # ,remainsonaveragethesamebetweenmodels A and B ,whichpointstothelatternotreallybeingstatisticallyfavored.The b g and M 1 / M min relationsalsodonotchangeappreciably.Nonetheless,the B tsdoprefer # log M values thatareveryclosetozero,acasethatisunphysicalsincethescatterofstellarmasses atxedhalomass(andvice-versa)isexpectedtobe > 0.15 ,asmentionedinSection 2.3.3 .Onaverage,thechangesinallotherttedparametersand n g are 20 %, whichsupportsthevalidityofthehalooccupationmodel.However,thisisnotan indicationthat A and B tsareequallyreliable.Fixingthenumberdensityplacesa strongconstraintontheHODmodel.TheinferreddensitiesfromEquation 26 do containsomeuncertaintysincetheyarepartiallyderivedfromthecontrolsample,butwe dobelievethatusingthemproducesamorephysicallyconsistentHODmodel.Thetted andderivedparametersfromthe B tsdonotfollowacontinuoustrendwithsample luminosity,butsufferfromdiscontinuousjumpsinsomecases.Thiswasalreadypointed outwhenusing n g -xed3-parametertsinSection 2.5.1 ,wherethisbehavioroccurred toalesserextentthanwith B .In A ,allvaluesdisplayasmoothandnearlycontinuous trendwithluminosity,whichmakesitmorereliablefromaphysicalstandpoint. 133

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WhenperformingatoftheSHMRwithresultsfrom B ,theerrorsbecome largeenoughtobeconsistentatabout1# levelwith A .Forexample,intheM13 parametrization,for A weobtained log M peak =12.43 0.08 =1.67 0.08 and 0 =0.59 0.03 ,whilefor B thesebecome log M peak =12.43 0.09 =2.14 0.98 and 0 =0.50 0.11 .WeobtainsimilarvalueswhenusingEGSasthereferencecatalog(see Appendix B ). 134

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REFERENCES AbbasU.etal.,2010,MNRAS,406,1306 AkaikeH.,1974,IEEETransactionsonAutomaticControl,19,716 AmblardA.etal.,2011,Nature,470,510 AshbyM.L.N.,StanfordS.A.,BrodwinM.,GonzalezA.H.,Martinez-MansoJ.,Bartlett J.G.,BensonB.A.,BleemL.E.,2013,ApJS,209,22 AugerM.W.,TreuT.,BoltonA.S.,GavazziR.,KoopmansL.V.E.,MarshallP.J., MoustakasL.A.,BurlesS.,2010,ApJ,724,511 AustermannJ.E.etal.,2012,inSocietyofPhoto-OpticalInstrumentationEngineers (SPIE)ConferenceSeries,Vol.8452,SocietyofPhoto-OpticalInstrumentation Engineers(SPIE)ConferenceSeries BaldryI.K.,GlazebrookK.,BrinkmannJ.,Ivezi c Z.,LuptonR.H.,NicholR.C.,Szalay A.S.,2004,ApJ,600,681 BarroG.etal.,2011a,ApJS,193,13 BarroG.etal.,2011b,ApJS,193,30 BehrooziP.S.,ConroyC.,WechslerR.H.,2010,ApJ,717,379 BehrooziP.S.,WechslerR.H.,ConroyC.,2013,ApJ,770,57 BellE.F.etal.,2004,ApJ,608,752 BenoistC.,MaurogordatoS.,daCostaL.N.,CappiA.,SchaefferR.,1996,ApJ,472, 452 BensonA.J.,BowerR.G.,FrenkC.S.,LaceyC.G.,BaughC.M.,ColeS.,2003,ApJ, 599,38 BerlindA.A.,WeinbergD.H.,2002,ApJ,575,587 BerlindA.A.etal.,2003,ApJ,593,1 BertinG.,CiottiL.,DelPrincipeM.,2002,A&A,386,149 B etherminM.,DoleH.,LagacheG.,LeBorgneD.,PeninA.,2011,A&A,529,A4 B etherminM.,WangL.,Dor eO.,LagacheG.,SargentM.,DaddiE.,CousinM.,Aussel H.,2013,A&A,557,A66 BeutlerF.etal.,2013,MNRAS,429,3604 BirnboimY.,DekelA.,2003,MNRAS,345,349 135

PAGE 136

BlakeC.,CollisterA.,LahavO.,2008,MNRAS,385,1257 BleemL.E.etal.,2012,ApJL,753,L9 BlumenthalG.R.,FaberS.M.,PrimackJ.R.,ReesM.J.,1984,Nature,311,517 BoltonA.S.,BurlesS.,KoopmansL.V.E.,TreuT.,GavazziR.,MoustakasL.A.,Wayth R.,SchlegelD.J.,2008,ApJ,682,964 BowerR.G.,BensonA.J.,MalbonR.,HellyJ.C.,FrenkC.S.,BaughC.M.,ColeS., LaceyC.G.,2006,MNRAS,370,645 BrodwinM.etal.,2006a,ApJ,651,791 BrodwinM.etal.,2008,ApJL,687,L65 BrodwinM.,LillyS.J.,PorcianiC.,McCrackenH.J.,LeF ` evreO.,FoucaudS.,Crampton D.,MellierY.,2006b,ApJS,162,20 BrownM.J.I.etal.,2008,ApJ,682,937 BruzualG.,CharlotS.,2003,MNRAS,344,1000 BuitragoF.,TrujilloI.,ConseliceC.J.,BouwensR.J.,DickinsonM.,YanH.,2008,ApJL, 687,L61 BullockJ.S.,KolattT.S.,SigadY.,SomervilleR.S.,KravtsovA.V.,KlypinA.A., PrimackJ.R.,DekelA.,2001,MNRAS,321,559 BundyK.etal.,2006,ApJ,651,120 CacciatoM.,vandenBoschF.C.,MoreS.,LiR.,MoH.J.,YangX.,2009,MNRAS,394, 929 CacciatoM.,vandenBoschF.C.,MoreS.,MoH.,YangX.,2013,MNRAS,430,767 CalabrettaM.R.,GreisenE.W.,2002,A&A,395,1077 CalzettiD.,ArmusL.,BohlinR.C.,KinneyA.L.,KoornneefJ.,Storchi-BergmannT., 2000,ApJ,533,682 CappellariM.etal.,2006,MNRAS,366,1126 CappellariM.etal.,2009,ApJL,704,L34 CappellariM.etal.,2007,MNRAS,379,418 CardielN.,1999,PhDthesis,,UniversidadComplutensedeMadrid,Spain,(1999) CarlstromJ.E.,AdeP.A.R.,AirdK.A.,BensonB.A.,Bleem,2011,PASP,123,568 CenR.,OstrikerJ.P.,1992,ApJL,399,L113 136

PAGE 137

CenarroA.J.,TrujilloI.,2009,ApJL,696,L43 CepaJ.,1998,,263,369 ChabrierG.,2003,PASP,115,763 CoilA.L.etal.,2008,ApJ,672,153 ColesP.,1993,MNRAS,262,1065 ConroyC.,GunnJ.E.,WhiteM.,2009,ApJ,699,486 ConroyC.,OstrikerJ.P.,2008,ApJ,681,151 ConroyC.,WechslerR.H.,2009,ApJ,696,620 ConroyC.,WechslerR.H.,KravtsovA.V.,2006,ApJ,647,201 CooperM.C.etal.,2006,MNRAS,370,198 CoorayA.,ShethR.,2002,Phys.Rep.,372,1 CouponJ.etal.,2012,A&A,542,A5 CowieL.L.,SongailaA.,HuE.M.,CohenJ.G.,1996,AJ,112,839 CrotonD.J.,GaoL.,WhiteS.D.M.,2007,MNRAS,374,1303 CrotonD.J.etal.,2006,MNRAS,365,11 DaddiE.etal.,2005,ApJ,626,680 DamjanovI.etal.,2009,ApJ,695,101 DavisM.etal.,2007,ApJL,660,L1 delaTorreS.etal.,2013,A&A,557,A54 DekelA.,BirnboimY.,2006,MNRAS,368,2 diSeregoAlighieriS.etal.,2005,A&A,442,125 DoleH.etal.,2006,A&A,451,417 DonosoE.,YanL.,SternD.,AssefR.J.,2013,ArXive-prints DuffyA.R.,SchayeJ.,KayS.T.,DallaVecchiaC.,2008,MNRAS,390,L64 DunkleyJ.,HlozekR.,SieversJ.,AcquavivaV.,AdeP.A.R.,AguirreP.,Amiri,2011, ApJ,739,52 DuttonA.A.,vandenBoschF.C.,2009,MNRAS,396,141 137

PAGE 138

EisensteinD.J.,HuW.,1999,ApJ,511,5 ElbazD.etal.,2007,A&A,468,33 FakhouriO.,MaC.-P.,2008,MNRAS,386,577 FakhouriO.,MaC.-P.,Boylan-KolchinM.,2010,MNRAS,406,2267 FallS.M.,EfstathiouG.,1980,MNRAS,193,189 Fern andezLorenzoM.,CepaJ.,BongiovanniA.,P erezGarc aA.M.,EderocliteA., Lara-L opezM.A.,Povi cM.,S anchez-PortalM.,2011,A&A,526,A72+ FiocM.,Rocca-VolmerangeB.,1997,A&A,326,950 FixsenD.J.,DwekE.,MatherJ.C.,BennettC.L.,ShaferR.A.,1998,ApJ,508,123 Foreman-MackeyD.,HoggD.W.,LangD.,GoodmanJ.,2013,PASP,125,306 FoucaudS.,ConseliceC.J.,HartleyW.G.,LaneK.P.,BamfordS.P.,AlmainiO.,Bundy K.,2010,MNRAS,406,147 FryJ.N.,1996,ApJL,461,L65 FryJ.N.,GaztanagaE.,1993,ApJ,413,447 GaoL.,NavarroJ.F.,ColeS.,FrenkC.S.,WhiteS.D.M.,SpringelV.,JenkinsA.,Neto A.F.,2008,MNRAS,387,536 GavazziR.,TreuT.,RhodesJ.D.,KoopmansL.V.E.,BoltonA.S.,BurlesS.,Massey R.J.,MoustakasL.A.,2007,ApJ,667,176 GeachJ.E.etal.,2013,ApJL,776,L41 GebhardtK.etal.,2003,ApJ,597,239 GeorgeE.M.,AdeP.,AirdK.A.,AustermannJ.E.,BeallJ.A.,Becker,2012,inSociety ofPhoto-OpticalInstrumentationEngineers(SPIE)ConferenceSeries,Vol.8452, SocietyofPhoto-OpticalInstrumentationEngineers(SPIE)ConferenceSeries Gonz alezV.,Labb eI.,BouwensR.J.,IllingworthG.,FranxM.,KriekM.,BrammerG.B., 2010,ApJ,713,115 G orskiK.M.,HivonE.,BandayA.J.,WandeltB.D.,HansenF.K.,ReineckeM., BartelmannM.,2005,ApJ,622,759 GrifnM.J.,SwinyardB.M.,VigrouxL.G.,2003,inSocietyofPhoto-Optical InstrumentationEngineers(SPIE)ConferenceSeries,Vol.4850,IRSpaceTelescopes andInstruments,MatherJ.C.,ed.,pp.686697 GrossanB.,SmootG.F.,2007,A&A,474,731 138

PAGE 139

GuoQ.etal.,2011,MNRAS,413,101 GuoQ.,WhiteS.,LiC.,Boylan-KolchinM.,2010,MNRAS,404,1111 HajianA.etal.,2012,ApJ,744,40 HallN.R.etal.,2010,ApJ,718,632 HamabeM.,KormendyJ.,1987,inIAUSymposium,Vol.127,StructureandDynamics ofEllipticalGalaxies,P.T.deZeeuw,ed.,pp.379+ HamblyN.C.etal.,2001,MNRAS,326,1279 HamiltonA.J.S.,1993,ApJ,417,19 HansenS.M.,SheldonE.S.,WechslerR.H.,KoesterB.P.,2009,ApJ,699,1333 HansonD.etal.,2013,PhysicalReviewLetters,111,141301 HartleyW.G.etal.,2013,MNRAS,431,3045 HenningJ.W.,AdeP.,AirdK.A.,AustermannJ.E.,BeallJ.A.,BeckerD.,Benson, 2012,inSocietyofPhoto-OpticalInstrumentationEngineers(SPIE)Conference Series,Vol.8452,SocietyofPhoto-OpticalInstrumentationEngineers(SPIE) ConferenceSeries HolderG.P.etal.,2013,ApJL,771,L16 HopkinsA.M.,BeacomJ.F.,2006,ApJ,651,142 HopkinsP.F.,BundyK.,HernquistL.,WuytsS.,CoxT.J.,2010,MNRAS,401,1099 HopkinsP.F.,HernquistL.,CoxT.J.,KeresD.,WuytsS.,2009,ApJ,691,1424 JrgensenI.,ChiboucasK.,FlintK.,BergmannM.,BarrJ.,DaviesR.,2006,ApJL,639, L9 JrgensenI.,ChiboucasK.,FlintK.,BergmannM.,BarrJ.,DaviesR.,2007,ApJL,654, L179 JorgensenI.,FranxM.,KjaergaardP.,1995,MNRAS,276,1341 JorgensenI.,FranxM.,KjaergaardP.,1996,MNRAS,280,167 JulloE.etal.,2012,ApJ,750,37 JuneauS.etal.,2005,ApJL,619,L135 KaiserN.,1984,ApJL,284,L9 KamionkowskiM.,KosowskyA.,StebbinsA.,1997,PhysicalReviewLetters,78,2058 139

PAGE 140

KatzN.,WeinbergD.H.,HernquistL.,1996,ApJS,105,19 KauffmannG.etal.,2003,MNRAS,341,33 KauffmannG.,NusserA.,SteinmetzM.,1997,MNRAS,286,795 KauffmannG.,WhiteS.D.M.,HeckmanT.M.,M enardB.,BrinchmannJ.,CharlotS., TremontiC.,BrinkmannJ.,2004,MNRAS,353,713 KennicuttR.C.etal.,2009,ApJ,703,1672 Kennicutt,Jr.R.C.,1998,,36,189 Kere sD.,KatzN.,FardalM.,Dav eR.,WeinbergD.H.,2009,MNRAS,395,160 Kere sD.,KatzN.,WeinbergD.H.,Dav eR.,2005,MNRAS,363,2 KlypinA.A.,Trujillo-GomezS.,PrimackJ.,2011,ApJ,740,102 KravtsovA.V.,BerlindA.A.,WechslerR.H.,KlypinA.A.,Gottl oberS.,AllgoodB., PrimackJ.R.,2004,ApJ,609,35 KroupaP.,2001,MNRAS,322,231 LagacheG.,BavouzetN.,Fernandez-CondeN.,PonthieuN.,RodetT.,DoleH., Miville-Desch enesM.-A.,PugetJ.-L.,2007,ApJL,665,L89 LagacheG.,HaffnerL.M.,ReynoldsR.J.,TufteS.L.,2000,A&A,354,247 LandyS.D.,SzalayA.S.,1993,ApJ,412,64 LeauthaudA.,TinkerJ.,BehrooziP.S.,BushaM.T.,WechslerR.H.,2011,ApJ,738,45 LeauthaudA.etal.,2012,ApJ,744,159 LevensonL.,MarsdenG.,ZemcovM.,AmblardA.,BlainA.,BockJ.,Chapin,2010, MNRAS,409,83 LewisA.,ChallinorA.,2006,Phys.Rep.,429,1 LewisA.,ChallinorA.,LasenbyA.,2000,ApJ,538,473 LimberD.N.,1953,ApJ,117,134 LinY.-T.,MohrJ.J.,2004,ApJ,617,879 LinY.-T.,MohrJ.J.,StanfordS.A.,2003,ApJ,591,749 LinderE.V.,2005,Phys.Rev.D,72,043529 LonghettiM.etal.,2007,MNRAS,374,614 MaC.-P.,FryJ.N.,2000,ApJ,543,503 140

PAGE 141

MagdisG.E.etal.,2012,ApJ,760,6 ManconeC.L.,GonzalezA.H.,2012,PASP,124,606 MandelbaumR.,SeljakU.,KauffmannG.,HirataC.M.,BrinkmannJ.,2006,MNRAS, 368,715 MarastonC.,2005,MNRAS,362,799 MarchesiniD.,vanDokkumP.G.,F orsterSchreiberN.M.,FranxM.,Labb eI.,WuytsS., 2009,ApJ,701,1765 MatsuokaY.,MasakiS.,KawaraK.,SugiyamaN.,2011,MNRAS,410,548 McBrideJ.,FakhouriO.,MaC.-P.,2009,MNRAS,398,1858 McCrackenH.J.,IlbertO.,MellierY.,BertinE.,GuzzoL.,ArnoutsS.,LeF ` evreO., ZamoraniG.,2008,A&A,479,321 McMahonJ.J.,AirdK.A.,BensonB.A.,BleemL.E.,BrittonJ.,CarlstromJ.E.,Chang, 2009,inAmericanInstituteofPhysicsConferenceSeries,Vol.1185,American InstituteofPhysicsConferenceSeries,YoungB.,CabreraB.,MillerA.,eds.,pp. 511514 MeneuxB.etal.,2009,A&A,505,463 MeneuxB.etal.,2008,A&A,478,299 MoH.J.,WhiteS.D.M.,1996,MNRAS,282,347 MoreS.,vandenBoschF.C.,CacciatoM.,MoH.J.,YangX.,LiR.,2009,MNRAS,392, 801 MoreS.,vandenBoschF.C.,CacciatoM.,SkibbaR.,MoH.J.,YangX.,2011,MNRAS, 410,210 MoscardiniL.,ColesP.,LucchinF.,MatarreseS.,1998,MNRAS,299,95 MostekN.,CoilA.L.,CooperM.,DavisM.,NewmanJ.A.,WeinerB.J.,2013,ApJ,767, 89 MosterB.P.,NaabT.,WhiteS.D.M.,2013,MNRAS,428,3121 MosterB.P.,SomervilleR.S.,MaulbetschC.,vandenBoschF.C.,Macci ` oA.V.,Naab T.,OserL.,2010,ApJ,710,903 MosterB.P.,SomervilleR.S.,NewmanJ.A.,RixH.-W.,2011,ApJ,731,113 MurrayN.,QuataertE.,ThompsonT.A.,2005,ApJ,618,569 MuzzinA.etal.,2013a,ApJS,206,8 141

PAGE 142

MuzzinA.,WilsonG.,DemarcoR.,LidmanC.,NantaisJ.,HoekstraH.,YeeH.K.C., RetturaA.,2013b,ApJ,767,39 NaabT.,JohanssonP.H.,OstrikerJ.P.,2009,ApJL,699,L178 NavarroJ.F.,FrenkC.S.,WhiteS.D.M.,1997,ApJ,490,493 NeisteinE.,vandenBoschF.C.,DekelA.,2006,MNRAS,372,933 NoeskeK.G.,WeinerB.J.,FaberS.M.,PapovichC.,KooD.C.,SomervilleR.S., BundyK.,2007,ApJL,660,L43 NorbergP.,BaughC.M.,Gazta nagaE.,CrotonD.J.,2009,MNRAS,396,19 NorbergP.etal.,2001,MNRAS,328,64 OkamotoT.,HuW.,2003,Phys.Rev.D,67,083002 PapovichC.,2008,ApJ,676,206 PeacockJ.A.,SmithR.E.,2000,MNRAS,318,1144 PeeblesP.J.E.,1980,Thelarge-scalestructureoftheuniverse PengC.Y.,HoL.C.,ImpeyC.D.,RixH.,2002,AJ,124,266 P erez-Gonz alezP.G.etal.,2008,ApJ,675,234 PetersonJ.R.,FabianA.C.,2006,Phys.Rep.,427,1 PhillippsS.,FongR.,FallR.S.E.S.M.,MacGillivrayH.T.,1978,MNRAS,182,673 PhlepsS.,PeacockJ.A.,MeisenheimerK.,WolfC.,2006,A&A,457,145 PilbrattG.L.etal.,2010,A&A,518,L1 PlanckCollaborationetal.,2013a,ArXive-prints PlanckCollaborationetal.,2013b,ArXive-prints PlanckCollaborationetal.,2013c,ArXive-prints PlanckCollaborationetal.,2011,A&A,536,A18 PressW.H.,SchechterP.,1974,ApJ,187,425 ReddickR.M.,WechslerR.H.,TinkerJ.L.,BehrooziP.S.,2013,ApJ,771,30 ReesM.J.,OstrikerJ.P.,1977,MNRAS,179,541 RossA.J.,PercivalW.J.,BrunnerR.J.,2010,MNRAS,407,420 S anchez-Bl azquezP.etal.,2006,MNRAS,371,703 142

PAGE 143

SayreJ.T.etal.,2012,inSocietyofPhoto-OpticalInstrumentationEngineers(SPIE) ConferenceSeries,Vol.8452,SocietyofPhoto-OpticalInstrumentationEngineers (SPIE)ConferenceSeries SchayeJ.etal.,2010,MNRAS,402,1536 ScoccimarroR.,ShethR.K.,HuiL.,JainB.,2001,ApJ,546,20 ScrantonR.etal.,2002,ApJ,579,48 SeljakU.,2000,MNRAS,318,203 ShangC.,HaimanZ.,KnoxL.,OhS.P.,2012,MNRAS,421,2832 ShenS.,MoH.J.,WhiteS.D.M.,BlantonM.R.,KauffmannG.,VogesW.,Brinkmann J.,CsabaiI.,2003,MNRAS,343,978 SherwinB.D.,DasS.,HajianA.,AddisonG.,BondJ.R.,Crichton,2012,Phys.Rev.D, 86,083006 SherwinB.D.,DunkleyJ.,DasS.,AppelJ.W.,BondJ.R.,CarvalhoC.S.,2011, PhysicalReviewLetters,107,021302 ShethR.K.,MoH.J.,TormenG.,2001,MNRAS,323,1 ShethR.K.,TormenG.,1999,MNRAS,308,119 ShirokoffE.etal.,2011,ApJ,736,61 SilkJ.,1977,ApJ,211,638 SimonP.,HetterscheidtM.,WolfC.,MeisenheimerK.,HildebrandtH.,SchneiderP., SchirmerM.,ErbenT.,2009,MNRAS,398,807 SkrutskieM.F.,CutriR.M.,StieningR.,WeinbergM.D.,SchneiderS.,CarpenterJ.M., Beichman,2006,AJ,131,1163 SmithR.E.etal.,2003,MNRAS,341,1311 SpringelV.,FrenkC.S.,WhiteS.D.M.,2006,Nature,440,1137 SpringelV.,HernquistL.,2003,MNRAS,339,289 SpringelV.etal.,2005,Nature,435,629 SpringelV.,YoshidaN.,WhiteS.D.M.,2001,,6,79 SternD.etal.,2005,ApJ,631,163 SwetzD.S.etal.,2011,ApJS,194,41 TegmarkM.etal.,2004,ApJ,606,702 143

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TegmarkM.etal.,2002,ApJ,571,191 TinkerJ.L.,LeauthaudA.,BundyK.,GeorgeM.R.,BehrooziP.,MasseyR.,RhodesJ., WechslerR.H.,2013,ApJ,778,93 TinkerJ.L.,RobertsonB.E.,KravtsovA.V.,KlypinA.,WarrenM.S.,YepesG., Gottl oberS.,2010,ApJ,724,878 TinkerJ.L.etal.,2012,ApJ,745,16 TinkerJ.L.,WeinbergD.H.,ZhengZ.,ZehaviI.,2005,ApJ,631,41 TinkerJ.L.,WetzelA.R.,2010,ApJ,719,88 TreuT.,EllisR.S.,LiaoT.X.,vanDokkumP.G.,2005,ApJL,622,L5 TrujilloI.,CenarroA.J.,deLorenzo-C aceresA.,VazdekisA.,delaRosaI.G.,CavaA., 2009,ApJL,692,L118 TrujilloI.,ConseliceC.J.,BundyK.,CooperM.C.,EisenhardtP.,EllisR.S.,2007, MNRAS,382,109 TrujilloI.,FerrerasI.,delaRosaI.G.,2011,ArXive-prints TrujilloI.etal.,2006,ApJ,650,18 Trujillo-GomezS.,KlypinA.,PrimackJ.,RomanowskyA.J.,2011,ApJ,742,16 VaccariM.,MarchettiL.,FranceschiniA.,AltieriB.,Amblard,2010,A&A,518,L20 ValeA.,OstrikerJ.P.,2006,MNRAS,371,1173 vandeVoortF.,SchayeJ.,BoothC.M.,DallaVecchiaC.,2011a,MNRAS,415,2782 vandeVoortF.,SchayeJ.,BoothC.M.,HaasM.R.,DallaVecchiaC.,2011b,MNRAS, 414,2458 vanderWelA.,FranxM.,vanDokkumP.G.,RixH.,IllingworthG.D.,RosatiP.,2005, ApJ,631,145 vanderWelA.,FranxM.,WuytsS.,vanDokkumP.G.,HuangJ.,RixH.,Illingworth G.D.,2006,ApJ,652,97 vanderWelA.,vanderMarelR.P.,2008,ApJ,684,260 vanDokkumP.G.etal.,2009,PASP,121,2 vanDokkumP.G.etal.,2010,ApJ,709,1018 vanEngelenA.etal.,2012,ApJ,756,142 144

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VazdekisA.,S anchez-Bl azquezP.,Falc on-BarrosoJ.,CenarroA.J.,BeasleyM.A., CardielN.,GorgasJ.,PeletierR.F.,2010,MNRAS,404,1639 VelanderM.,KuijkenK.,SchrabbackT.,2011,MNRAS,412,2665 VieroM.P.etal.,2009,ApJ,707,1766 VieroM.P.,MoncelsiL.,QuadriR.F.,ArumugamV.,AssefR.J.,B ethermin,2013a, ApJ,779,32 VieroM.P.,WangL.,ZemcovM.,AddisonG.,AmblardA.,ArumugamV.,Aussel, 2013b,ApJ,772,77 VogelsbergerM.,GenelS.,SijackiD.,TorreyP.,SpringelV.,HernquistL.,2013, MNRAS,436,3031 WakeD.A.etal.,2011,ApJ,728,46 WangL.etal.,2013,MNRAS,431,648 WatsonD.F.,BerlindA.A.,ZentnerA.R.,2011,ApJ,738,22 WechslerR.H.,BullockJ.S.,PrimackJ.R.,KravtsovA.V.,DekelA.,2002,ApJ,568,52 WeinbergD.H.,MortonsonM.J.,EisensteinD.J.,HirataC.,RiessA.G.,RozoE.,2013, Phys.Rep.,530,87 WeinbergN.N.,KamionkowskiM.,2003,MNRAS,341,251 WetzelA.R.,CohnJ.D.,WhiteM.,2009,MNRAS,395,1376 WetzelA.R.,TinkerJ.L.,ConroyC.,vandenBoschF.C.,2013,MNRAS,432,336 WhiteS.D.M.,DavisM.,EfstathiouG.,FrenkC.S.,1987,Nature,330,451 WhiteS.D.M.,FrenkC.S.,1991,ApJ,379,52 WhiteS.D.M.,ReesM.J.,1978,MNRAS,183,341 YangX.,MoH.J.,vandenBoschF.C.,2003,MNRAS,339,1057 YangX.,MoH.J.,vandenBoschF.C.,ZhangY.,HanJ.,2012,ApJ,752,41 ZahidH.J.,KewleyL.J.,BresolinF.,2011,ApJ,730,137 ZehaviI.,PatiriS.,ZhengZ.,2012,ApJ,746,145 ZehaviI.etal.,2011,ApJ,736,59 ZehaviI.etal.,2005,ApJ,630,1 145

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ZentnerA.R.,BerlindA.A.,BullockJ.S.,KravtsovA.V.,WechslerR.H.,2005,ApJ, 624,505 ZhengZ.,2004,ApJ,610,61 ZhengZ.etal.,2005,ApJ,633,791 ZhengZ.,CoilA.L.,ZehaviI.,2007,ApJ,667,760 ZuY.,ZhengZ.,ZhuG.,JingY.P.,2008,ApJ,686,41 146

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BIOGRAPHICALSKETCH JesusMartinez-Mansoearnedhisbachelor'sdegreeinphysicsattheUniversity ofSevilla,Spain.Duringthattime,hespentayearabroadattheLudwigMaximilian UniversityofMunich,wherethegreatscienticatmospheremadehimdecideto pursueaPhDafterwards.ThisbecamearealitywhenheobtainedtheDoctoralAlumni FellowshipfromtheUniversityofFlorida.HisrsttwoyearsintheDepartmentof AstronomywerespentworkingunderthesupervisionofRafaelGuzmanonthestudy ofstellarpopulationsanddynamicalmassesofgalaxiesthroughopticalspectroscopy. Then,underthetutelageofAnthonyGonzalez,hefocusedhisthesisonthestatistical studyofthehighredshiftgalaxypopulationandhowitspropertiesaredeterminedbythe underlyingdistributionofdarkmatter. 147