Mon.Not.R.Astron.Soc.000,1–6(2015) Printed3August2015 (MNLATEXstylefilev2.2) The bias of the submillimetre galaxy population: SMGs are poor tracers of the most massive structures in the z 2 Universe ∼ Tim B. Miller1(cid:63), Christopher C. Hayward2,3 , Scott C. Chapman1 and Peter S. † Behroozi4 ‡ 1DepartmentofPhysicsandAtmosphericScience,DalhousieUniversity,6310CoburgRoad,Halifax,NSB3H4R2,Canada 2TAPIR350-17,CaliforniaInstituteofTechnology,1200E.CaliforniaBoulevard,Pasadena,CA91125,USA 3Harvard–SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA 4SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA 5 1 0 2 AcceptedforpublicationinMNRAS l u J ABSTRACT 1 It is often claimed that overdensities of (or even individual bright) submillimetre-selected 3 galaxies (SMGs) trace the assembly of the most-massive dark matter structures in the Uni- verse. We test this claim by performing a counts-in-cells analysis of mock SMG catalogues ] derivedfromtheBolshoicosmologicalsimulationtoinvestigatehowwellSMGassociations A trace the underlying dark matter structure. We find that SMGs exhibit a relatively complex G bias: some regions of high SMG overdensity are underdense in terms of dark matter mass, . and some regions of high dark matter overdensity contain no SMGs. Because of their rar- h ity, Poisson noise causes scatter in the SMG overdensity at fixed dark matter overdensity. p Consequently,richassociationsofless-luminous,more-abundantgalaxies(i.e.Lyman-break - o galaxyanalogues)tracethehighestdarkmatteroverdensitiesmuchbetterthanSMGs.Even r on average,SMG associationsare relatively poortracers of themost significant darkmatter st overdensities because of ‘downsizing’: at z (cid:46) 2.5, the most-massive galaxies that reside in a thehighestdarkmatteroverdensitieshavealreadyhadtheirstarformationquenchedandare [ thusnolongerSMGs.Atagivenredshift,ofthe10percentmost-massiveoverdensities,only 2 25percentcontainatleastoneSMG,andlessthanafewpercentcontainmorethanone ∼ v SMG. 5 Key words: cosmology: theory – cosmology: large-scale structure of Universe – galaxies: 0 1 clusters:general–galaxies:high-redshift–methods:numerical–submillimeter:galaxies. 4 0 . 1 0 1 INTRODUCTION the highest-redshift proto-clusters (e.g. Capak et al. 2011; Walter 5 etal.2012).Thus,observationsofbrightSMGsshouldprobetheir 1 Submillimetre-selectedgalaxies(SMGs;seeCaseyetal.2014for environment and trace significant overdensities that can be inter- iv: a5×re1c0en12trLe(cid:12)vi,erwep),rewseitnhtttyhpeircaarlesintfarnadremdo(IsRte)xlturmemineoesxitaimespolefsLoIfRst(cid:38)ar pSrMetGedsisnetthcerictiocnatlecxotnosftlraarignets-socnalceosstmruocltougreicsailmmuloadtieolsn.s.Therefore, X forminggalaxies.TheLIRofanSMGimpliesanimmensestarfor- r mationrate,typicallySFR∼500−1000M(cid:12)yr−1,assumingthat InterestinSMGassociationshasgrowninrecentyearsasin- a thereisnotasignificantcontributiontoLIRfromdeeplyobscured creasing numbers of SMG associations have been detected (e.g. AGN. SMGs allow us to probe the mechanisms behind most in- Blainetal.2004;Chapmanetal.2005;Chapmanetal.2009;Geach tense star formation events in the Universe and can elucidate the etal.2005;Daddietal.2009;Dannerbaueretal.2014;MacKen- highestSFRssustainableinagalaxy.Becauseoftheirextremena- zie et al. 2014; Smail et al. 2014). Furthermore, Clements et al. ture,SMGsprovidelaboratoriestotestthelimitsofhydrodynam- (2014)havedemonstratedthatsomePlancksourcestraceoverden- icalsimulationsofgalaxies(e.g.Narayananetal.2010;Hayward sitiesofdustystar-forminggalaxies,andtheysuggestedthatsuch etal.2011;Hayward2013). observations can be used to investigate the epoch of galaxy clus- Massivestarburstgalaxiesappeartogrowinthemostmassive ter formation. However, this claim relies on the assumption that halos(Hickoxetal.2012),thusmakingthempotentialtracersfor overdensitiesofdustystar-forminggalaxiescorrespondtogalaxy clustersintheprocessofformation. (cid:63) E-mail:[email protected] Thereissomeobservationalevidencethatcallsthisclaiminto † MoorePrizePostdoctoralScholarinTheoreticalAstrophysics question: in their study of the GOODS-N field, Chapman et al. ‡ GiacconiFellow (2009)foundanassociationof8SMGsatz ≈ 1.99.Theassoci- (cid:13)c 2015RAS 2 Miller,Hayward,Chapman&Behroozi atedstructurewasonlyatypicaloverdenseregion,asindicatedby thewell-sampledopticalspectroscopyinthisregion,thatwouldnot 8 SMG formavirializedclusterbyz = 0.Moreover,Blainetal.(2004) 6 RandomLBG foofuenvdoltvheadt t‘heextrcelumsetelyrinrgedleonbgjethctso’f(SEMROGss)iastczon∼sist1enatnwdizth=tha0t alaxy 4 LBG g clusters,whichwouldsuggestthatthedescendantsofSMGswould δ 2 tendtobefoundinrichclusterenvironments;however,thisinter- 0 pretation implies a comoving space density of clusters that is at 50 least an order of magnitude greater than that observed. These re- 40 sultssuggestthatperhapsassociationsofSMGstraceparticularly active phases in relatively modest-mass overdensities rather than galaxy 2300 thehighestoverdensitiesandthushavearelativelycomplexclus- δ 10 teringbias. 0 To test this possibility, we have performed a counts-in-cells analysis on the Hayward et al. (2013a, hereafter H13) simulated 2 0 2 4 6 8 10 − SMGcataloguestoinvestigatetherelationshipsofSMGsandmore δmass modestlystar-forminggalaxiestotheunderlyingdarkmatterstruc- ture.WefirstinvestigatetheclusteringbiasesofSMGsandLyman- Figure1.Top:OverdensityofLBGs(0.1mJy<S850 <1mJy)vs.over- break-galaxy(LBG)analogues.Wethenstudythepropertiesofin- densityofdarkmatter,δmass,fortherandomcells.Thereisaclearcorrela- dividualassociationsofSMGsandLBGanalogues. tionbetweenδgalaxyandδmass,whichindicatesthattheclusteringofthe LBGstracestheclusteringofthedarkmatterwell.Bottom:Similartothe toppanel,butforSMGs(S850 >3mJy;redcircles)andarandomsubset ofLBGsselectedtohavenumberdensityequaltothatoftheSMGs(blue 2 METHODS diamonds).(TheSMGpointshavebeenshiftedupwardsby5forclarity.) ToanalyzethebiasintheSMGpopulation,weusethemockSMG TheSMGsandrandomsubsetofLBGsexhibitalargescatterinδgalaxy atagivenδmass.ThisresultindicatesthatbecauseoftherarityofSMGs, cataloguesofH13,whichweregeneratedbyassigninggalaxyprop- PoissonnoisecausesSMGoverdensitiestobepoortracersofdarkmatter ertiestodarkmatterhaloesfromacosmologicalcollisionlessdark overdensities. mattersimulationusingsubhaloabundancematchingandthenas- signingsubmmfluxdensitiesusingafittingfunctionderivedfrom the results of performing radiative transfer on idealized hydrody- mJy is ∼ 140 M(cid:12)yr−1) and those with 0.1 < S850 < 1 namical simulations. We will summarize the H13 methodology mJy as LBGs (this range corresponds to median SFR values of here,andwereferthereadertoH13forfulldetails. ∼10−50M(cid:12)yr−1).WestudythebiasofSMGsandLBGsand Using halo catalogues from the Bolshoi simulation (Klypin, identifySMGandLBGassociations(orredshiftspikes;e.g.Chap- Trujillo-Gomez, & Primack 2011; Behroozi et al. 2013b,c), we man et al. 2005; Chapman et al. 2009) using a simple counts-in- constructedmocklightconesbystartingateightrandomlocations cellsanalysis(e.g.Adelbergeretal.1998).Specifically,wedivide within the simulation and selecting haloes along a randomly ori- eachofthe8mockcataloguesintocellswithangulardimensions ented sightline with an 84’ x 84’ (1.96 deg2) field of view from 10arcmin×10arcminanddepthdz=0.05;theresultsaresimilar z = 0.5toz = 8.Wecalculatedcosmologicalredshifts,includ- ifweusecellswithsidelengthsequaltotwicethesevalues.Weuse ingtheeffectsofhalopeculiarvelocities.Wethenassignedstellar asubsetof10,000ofthesecellsforcalculatingtheclusteringbias masses(M(cid:63))andSFRsusingtheredshift-dependentstellarmass– and for making comparisons to the properties of SMG and LBG halomassandSFR–halomassrelationsofBehroozi,Wechsler,& associations;werefertothesecellsas‘randomcells’.Toidentify Conroy(2013a),whichincludescatteratfixedhalomassandred- associations,westartwiththesamecells.However,toensurethat shift.Weincludedasimplemodelforsatellitequenching:satellite we do not divide potential associations by using a fixed grid, we SFRswerereducedbyafactorequaltotheircurrentsubhalomass shiftthecellsby1-arcminintervals10timesanddefineanSMG dividedbythepeakmassintheirsubhalo’smassaccretionhistory. (LBG)associationasthegalaxiescontainedinthecellthatcontains Weassigneddustmasses(Md)tothehaloesusingtheempirically themaximumnumberofSMGs(LBGs).Weensurethatwedonot basedmethodofHaywardetal.(2013b).Finally,weassigned850- countasingleassociationmultipletimes.Wecalculatetotaldark µmfluxdensities(S850)usingthefollowingfittingfunction,which mattermassesforeachcellbysummingthedarkmattermassesof wasderivedbasedontheresultsofperformingdustradiativetrans- allhaloesofmass> 1010 M(cid:12) (becausetheBolshoisimulationis fer on hydrodynamical simulations of idealized disc galaxies and incompletebelowthishalomass)containedinthecell. mergers(Haywardetal.2011,2013b): As discussed in detail in H13, the H13 model does not in- (cid:18) (cid:19)0.43(cid:18) (cid:19)0.54 clude the effect of starbursts (i.e. the extended tail to high SFR SFR M S850 =0.81mJy 100 M yr−1 108Md , (1) at a given stellar mass and redshift). Because one would expect (cid:12) (cid:12) thatinteraction-inducedstarburstswouldmostaffecttheSFRsof whereweincorporatedthescatterintherelationof0.13dex(Hay- SMGsinhighlyoverdenseregionsandthuspotentiallyalterourre- ward et al. 2011) when assigning S850. Note that because S850 sults,wehaveextendedtheH13modelbyincludingamodelfor scalessublinearlywithbothSFRandMd,thepredictedS850 val- interaction-inducedstarbursts.Foreachgalaxy,wecheckwhether uesarerelativelyinsensitivetothemodeldetails.Furthermore,the ithasaneighboringgalaxywithstellarmasswithinafactorof3of S850–M(cid:63) relation predicted in this manner agrees well with that itsown(suchthatthepairwouldconstitutea‘major’merger)that observed(Daviesetal.2013). is located within a physical distance of dweak. If so, the SFRs of Throughoutthiswork,werefertomockgalaxieswithS850 > bothgalaxiesareboostedbyafactorofbweak.Iftheseparationis 3 mJy as SMGs (the median SFR for sources with S850 ∼ 3 lessthandstrong < dweak,weinsteadboosttheSFRsbyalarger (cid:13)c 2015RAS,MNRAS000,1–6 SMGsastracersoflarge-scalestructure 3 10 10 Mass=1.9z9=⇥21.30914(M�/h) SMGs Mass=8.6z1=×21.30313(M(cid:12)/h) SMGs 5 5 n) n) mi mi c c ar 0 ar 0 ( ( C C E E D D 5 5 � − 10 10 � 10 5 0 5 10 − 10 5 0 5 10 � � − − RA(arcmin) RA(arcmin) Figure2.Spatialdistributionsofgalaxiesnearthe2richestSMGassociations(whicheachcontain6SMGs;thecellsaremarkedwithdashedlines).The SMGs(LBGs)aredenotedwithred(blue)points,thesizesofwhichareproportionaltoS850.Thespatialdistributionsofthegalaxiesreflectthefilamentary structureofthedarkmatterdistribution. factor,bstrong >bweak.Weexperimentedwithdifferentreasonable Table1.DemographicsofSMGassociations parametervalues,asjudgedbasedontheresultsofidealizedhydro- dynamicalsimulationsofmergers(e.g.Coxetal.2008;Torreyetal. NSMGa Percentageb Massc Separationd 2012;Haywardetal.2014),andfoundthattheresultswerequal- (1013M(cid:12)) (Mpc) itativelyunaffectedevenfortheextremescenarioofdweak = 15 kpc,bweak =10,dstrong =5kpc,anddstrong =100.However,it 1 64 5.9±3.0 – 2 23 7.7±3.6 7.6 isimportanttonotethatthecataloguesareincompleteformergers 3 10 9.1±4.7 6.1 withsmallseparations(Behroozietal.2013b),whichwouldresult 4 2 10.3±2.8 5.2 inanunderestimateofthenumberofinteractinggalaxies.Never- 5 0.8 12.6±2.6 4.3 theless, this incompleteness likely does not affect our results be- 6 0.2 14.1±5.6 4.6 causealthoughinteractionscouldboostthesubmmfluxesofsome galaxiesandincreasetheclusteringsignalon(cid:46)10kpcscales,the aNumberofSMGsinacell. bPercentageofSMGsinsuchas- clusteringonlargerscalesshouldbeunaffected.Inallfigures,we sociations. cMedianmassofsuchassociationsat2<z <3. showtheresultsfortheoriginalH13model,butthecorresponding dMedianpairwiseseparationsofSMGsinsuchassociations. plotsfortheboostedmodelsaresimilar. blendNinogteotfhamtuulntilpikleegHa1la3x,iweseihnatvoeanositnignlceosrpuobrmatmedstohuerceeffeinctsthoisf δmass = MDM<−M<DMM>DM >, (3) work,althoughtheoreticalarguments(Haywardetal.2011,2012, whereMDMisthemassofdarkmatterinacelland<MDM >is 2013b;H13)andobservations(e.g.Karimetal.2013;Hodgeetal. themeandarkmattermasspercell. 2013) suggest that blending significantly affects the single-dish- Fig.1showstheoverdensityδgalaxyofSMGs(S850 >3mJy; detectedSMGpopulation.Thereasonisthatthesizesoftheassoci- redcircles),LBGs(0.1mJy<S850 <1mJy;blackdiamonds)and ationsaremuchgreaterthanthebeamsizesofsingle-dishsubmm arandomsubsetofLBGsselectedtohavenumberdensityequalto telescopes (see below). Thus, although the detailed results could thatoftheSMGs(bluediamonds)vs.overdensityofdarkmatter, be affected by blending, our conclusions would be unchanged if δmass,fortherandomcells.ForthetotalLBGpopulation,δgalaxy blendingwereincorporated.Furthermore,wewishtoanalyzehow and δmass are tightly correlated, which indicates that LBG over- well individual submm-bright galaxies, which would be resolved densitiesaregoodtracersofdarkmatteroverdensities.Theslope by e.g. the Atacama Large Millimeter Array, rather than blended ofthebest-fittinglinearrelation(i.e.thebias,b ≡ δgalaxy/δmass) submmsources(whichdependonthebeamsizeoftheinstrument is0.98±0.01,andthemeansquarederror(MSE)is0.1.Forthe usedtodetectthemandarethusalessgeneralpopulationthanre- SMGsandrandomsubsetofLBGs,thereisacorrelationbetween solvedsources)tracedarkmatterstructures. δgalaxyandδmass,butitexhibitssignificantscatter.Thebiasvalues are1.3±0.1and1.0±0.1fortheSMGsandLBGs,respectively, andtheMSEvaluesare17and16.ThefactthattheSMGsandran- domLBGsexhibitsimilarscatterindicatesthatPoissonnoisedue 3 RESULTS to the rarity of SMGs is the reason for the complicated relation- For each cell, we calculate the number overdensity of SMGs shipbetweenδgalaxyandδmassforthispopulation.Thus,although (δSMG)andLBGs(δLBG)usingthefollowingequation: SMGsareslightlymorebiasedthanLBGs,SMGoverdensitiesare poortracersoftheunderlyingdarkmatteroverdensities.Thisef- N −<N > δgalaxy = gala<xyN ga>laxy , (2) fectmayexplaintheresultsofBlainetal.(2004)discussedabove. galaxy We now investigate the properties of individual SMG and whereNgalaxyisthenumberofgalaxiesinacelland<Ngalaxy > LBG redshift associations in detail. Table 1 presents the fraction isthemeannumberofgalaxiespercell.Wealsocalculatethedark ofSMGsinassociationsandthemedianseparationoftheSMGs mattermassoverdensityofeachcell, inthedifferenttypesofassociations.TypicalSMGsarenotinas- (cid:13)c 2015RAS,MNRAS000,1–6 4 Miller,Hayward,Chapman&Behroozi 15 10 Random Random N 150 ≥≥100519LS5MBLGGBssGs N 5 134-SS2MMSMGGssGs 5SMGs 1015 1015 6SMGs h) h) / / (cid:12) � M M ( ( s s s s a a m 1014 m 1014 er er att att m m k k ar ar D D 1013 1013 1.5 2.0 2.5 3.0 3.5 3 6 9 12 1.5 2.0 2.5 3.0 3.5 5 10 z N z N Figure3.Left:totaldarkmattermassinacellversusredshiftofthecellforcellsthatcontain(cid:62)5SMGs(redcircles),cellswith(cid:62)195LBGs(thisnumberwas selectedtoyieldthe10richestLBGassociations;blackcircles),cellsthatcontainexactly100LBGs(greencircles)andrandomlyselectedcells(bluepoints). Theredshiftanddarkmattermassdistributionsareshownnexttotherespectiveaxes.ComparedwiththerichestLBGassociations,theSMGassociations traceless-massive,higher-redshiftstructures.Associationsof100LBGstracelower-massdarkmattersubstructuresthatspanthefullredshiftrangeconsidered. Right:similartotheleftpanel,butwithcellsclassifiedaccordingtothenumberofSMGsthattheycontain.CellswithlowernumbersofSMGstendtoinclude lessdarkmatter.Notably,someofthemostoverdenseregionscontainnoSMGs. sociations,butasubstantialminority(36percent)are.Onlyafew thanthevaluesfortheSMGassociations,whereastherichestLBG percentofSMGsarelocatedinrichassociationsoffourormore associations more faithfully trace the cells with the largest dark SMGs. The richer associations exhibit lower median separations, mattermasses. whichsuggeststhattheseassociationscorrespondtohigherover- Forcomparison,weshowmore-modestLBGassociationsthat densities(whichwillbeconfirmedbelow).Fig.2showsthespatial containexactly100LBGs(greencircles).Asexpected,theseLBG distributions of the SMGs and LBGs in two mock SMG associ- associationstracelessmassivesubstructuresthantherichestLBG ations.Theassociationseachcontain6SMGs.Thespatialdistri- associations.Themediandarkmattermassofthe100-LBGassoci- butionsofthegalaxiesexhibitclearfilamentarystructures,which ationsissimilartothatoftheSMGassociations,9.1×1013 M(cid:12), reflecttheunderlyingstructureofthe‘cosmicweb’.Notethatin- butthe100-LBGassociationsspanabroaderrangeofredshifts. corporatingblending(usingatypicalbeamof(cid:46)30arcsec)would The right panel of Fig. 3 shows the total dark matter mass tendtoincreasethesubmmfluxesoftheLBGsandblendtheSMGs in a cell versus redshift of the cell for cells that contain one or withLBGs,butitwouldnotblendanyofthebrightSMGs.Thus,a moreSMGs(colouredaccordingthenumberofSMGs).Thisfigure blendedversionwouldpotentiallycontainadditionalbrightSMGs demonstratesmultipleinterestingresults:first,manyofthehighest and therefore be comparable to observed SMG associations (e.g. darkmatteroverdensitiesatagivenredshiftcontainnoSMGs(the Chapmanetal.2009;Dannerbaueretal.2014). bluepointswithdarkmattermass(cid:38)2×1014M(cid:12)).Atagivenred- OurgoalistounderstandhowwellSMGassociationstracethe shift,ofthe10percentmost-massiveoverdensities,only∼25per highestdarkmatteroverdensities.Todoso,itisinstructivetocom- centcontainatleastoneSMG,andlessthanafewpercentcontain parethetotaldarkmattermassinindividualcellsthatareidentified morethanoneSMG.Consequently,findingdarkmatteroverdensi- asSMGorLBGassociationswiththevaluesforrandomlyselected tiesusingSMGsassignpostswillcauseonetomissmanyofthe cells.IfSMGassociationstracethemostsignificantoverdensities, highestoverdensities.Second,cellswithlowernumbersofSMGs thesecellsshouldcontainmoredarkmattermassthanothercells tendtohavelessdarkmatter.Finally,theminimummassnecessary at a given redshift. The left panel of Fig. 3 shows the total dark foracelltohostanSMGis∼1013M(cid:12),whichisconsistentwith matter in a given cell versus the redshift of the cell for cells that theresultsofinferencesfromtheclusteringofrealSMGs(Hickox contain(cid:62) 5SMGs(redcircles),the10richestLBGassociations etal.2012). ((cid:62)195LBGsinacell;blackcircles),cellsthatcontainexactly100 LBGs(greencircles),andasubsetofrandomlyselectedcells(blue points).Theredshiftanddarkmattermassdistributionsareshown 4 SUMMARYANDDISCUSSION nexttotherespectiveaxes. ItisimmediatelyclearthattheSMGassociationsdonottrace WehaveusedmockSMGcataloguestodemonstratethatSMGas- the most significant dark matter overdensities, although they do sociationsarepoortracersofthehighestoverdensitiesofdarkmat- tracerelativelyhighoverdensities.ComparedwiththerichestLBG ter. At higher redshifts (z (cid:38) 2.5), the richest SMG associations associations, which should be considered analogous to observed tracesomeofthehighestoverdensitiesbecausethemost-massive associations (i.e. redshift spikes) of LBGs, the SMG associations galaxiesinthoseregionsarestillformingstarsrapidly.However, tendtohavelowerdarkmattermasses(amedianof1.2×1014M(cid:12) such associations are rare, and the majority of the highest over- fortheSMGassociationscomparedwith2.2×1014 M(cid:12) forthe densitiesdonothostasingleSMG,letaloneanSMGassociation. (cid:62) 195−LBGassociations)andarelocatedathigherredshifts(the Consequently,SMGassociationsarehighlyincompletetracersof medianvaluesfortheSMGandLBGassociationsare2.4and2.0, thehighestoverdensitiesevenatz (cid:38) 2.5.Thesituationisworse respectively).Furthermore,therearemanyrandomlyselectedcells at z (cid:46) 2.5: many of the most-massive galaxies, which reside in thathavedarkmattermassesthatarecomparabletoorevengreater the highest dark matter overdensities, have already had their star (cid:13)c 2015RAS,MNRAS000,1–6 SMGsastracersoflarge-scalestructure 5 formationquenched.(Independentlyofredshift,thehaloswiththe anisotropies Cowley et al. (2015b) than did the previous model, highest ratio of SFR to halo mass are those with halo masses of whichsuggests(perhapsunsurprisingly)thattheCIBmayprovide ∼ 1012 M(cid:12) at that redshift; e.g. Behroozi et al. 2013a; Moster moreinsightintothenatureofSMGsthanclusteringdoes. et al. 2013; Sparre et al. 2015.) Consequently, the z (cid:46) 2.5 dark Granatoetal.(2014)performeddustradiativetransferonsim- matteroverdensitiesarelesslikelytocontainSMGs. ulated (proto-)clusters at z (cid:38) 1 to determine whether they could In our model, galaxy SFRs are assigned using a redshift- reproducethepropertiesofthePlanck sourcesofClementsetal. dependent SFR–halo mass relation and a model for satellite (2014).Theyfoundthattheirsimulated(proto-)clustershadlower quenching. The parameters of the model are constrained by fit- totalSFRsthaninferredfromobservations.Thistensionsuggests ting to a wide range of observations (Behroozi et al. 2013a). that associations of dusty galaxies should be investigated further Consequently, the fact that some fraction of massive galaxies are bothintermsoftheoryandobservation. quenched even at z ∼ 2 is not a prediction. The utility of our It is worthwhile to clarify why we have claimed that SMGs model is that it can be used to determine the consequences of are ‘poor tracers’ of high overdensities. In our model, the (cid:62) 5- quenching/downsizing for the clustering of the SMG population. SMGassociationseachlieinoneofthemost-massivecellsattheir Furthermore, because we determine the submm flux densities of respectiveredshifts.Inthissense,SMGassociationsdotracehigh ourgalaxiesself-consistentlyusingafittingfunctionderivedfrom overdensities.However,suchassociationsareuncommon(thereare radiative transfer calculations, there is not a monotonic mapping 11inour∼16−deg2mockcatalogue),whichisconsistentwiththe betweenSFRandsubmmfluxdensity(agalaxywitharelatively factthatonlyafewoverdensitiesof(cid:62) 5brightSMGshavebeen modest SFR can still be submm-bright if it has sufficiently high reportedintheliterature(Tamuraetal.2009;Chapmanetal.2009; dust mass). Thus, the results are specific to the SMG population Dannerbaueretal.2009;Maetal.2015).Moreover,atz (cid:46) 2.5, rather than just the most rapidly star-forming galaxies (cf. Dave´ almostnoneofthehighestoverdensitiescontain>3SMGsdespite etal.2010).Finally,ourmodelexplicitlyaccountsforthestochas- alargefractionoftheSMGpopulationbeinglocatedatz (cid:46) 2.5. ticitythatisinherentintheSMGselectionbecausebrightSMGsare Thus,searchingforhighoverdensitiesusingSMGassociationsas anextremepopulation;thus,onemayselectagalaxyasanSMG beaconswouldresultinaveryincompletesampleatbest. because it is in a short-lived phase of elevated SFR (perhaps due Overall,ourresultsurgecautionwheninterpretingSMGas- toaninteraction)orbecauseithasanespeciallyhighsubmmflux sociationsinthecontextoflarge-scalestructure.Becauseoftheir densityforitsSFRanddustmass(becauseofe.g.anespeciallyex- rarity,PoissonnoisecausessignificantscatterintheSMGoverden- tendedgeometry).GN20couldbeareal-Universeexampleofthe sityatfixeddarkmatteroverdensity(i.e.SMGsstochasticallysam- latter.Consequently,Poissonnoisecontributestothescatterinthe plethehighestoverdensities).Consequently,althoughthehighest- valueofδSMG atagivenδmass andcausessomeofthemostsig- redshift SMG associations trace some of the highest dark matter nificantoverdensitiestocontainfewornobrightSMGs.Moreover, overdensities at those redshifts, most of the highest overdensities ourmodelsuggeststhatsomethebrightestSMGsintheUniverse do not host SMG associations. At lower redshifts (z (cid:46) 2.5), the maylieinrelativelyisolatedmassivedarkmatterhalos,consistent situationisworse:thehighestoverdensitiestendtocontainonlya withobservationalfindings(Chapmanetal.2015). fewSMGsatmost,andthemajoritydonotcontainasingleSMG. A few other theoretical works have investigated the cluster- Thus,ifonewishestoidentifyprotoclusters,thecomplicatedbias ingoftheSMGpopulation.Dave´ etal.(2010)studiedtheproper- ofSMGsmakesthemless-than-idealbeacons. tiesofthemostrapidlystar-forminggalaxies,whichtheyconsid- ered SMG analogues, in a cosmological hydrodynamical simula- tion.Becauseofthetight,monotonicSFR–stellarmassrelationfor star-forminggalaxiesintheirsimulation,theyeffectivelyselected ACKNOWLEDGEMENTS themost-massivestar-forminggalaxiesintheirsimulation.Conse- We thank Neal Katz for useful discussion and Phil Hopkins quently, they found that their simulated SMGs were highly clus- teredandbiased,withacorrelationlengthr0 ≈ 10h−1 comoving forcommentsonthemanuscript.Wethanktheanonymousreferee foraconstructivereportthathelpedimprovethemanuscript.CCH Mpcandbiasof∼6. isgratefultotheGordonandBettyMooreFoundationforfinancial Cowleyetal.(2015b)analyzedtheclusteringofSMGsinthe supportandacknowledgesthehospitalityoftheAspenCenterfor currentDurhamsemi-analyticalmodel(Cowleyetal.2015a;Lacey Physics, which is supported by the National Science Foundation etal.,inpreparation).Inthismodel,bright(S850 >4mJy)SMGs atz ∼ 2.5exhibitacorrelationlengthofr0 = 5.5h−1 Mpcand Grant No.PHY-1066293. PSB was supportedby a Giacconi Fel- lowship provided through the Space Telescope Science Institute, a bias of ∼ 2.5. 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