Dark Energy Survey Year 3 Results: clustering redshifts – calibration of the weak lensing source redshift distributions with redMaGiC and

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Dark Energy Survey Year 3 Results: clustering redshifts – calibration of the weak lensing source redshift distributions with redMaGiC and

BOSS/eBOSS

M. Gatti,

¹^‹

G. Giannini ,

¹^‹

G. M. Bernstein ,

²

A. Alarcon,

³

J. Myles ,

⁴

A. Amon,

⁵

R. Cawthon ,

⁶

M. Troxel,

⁷

J. DeRose,

⁸^,⁹

S. Everett,

⁹

A. J. Ross ,

¹⁰

E. S. Rykoff,

⁵^,¹¹

J. Elvin-Poole,

¹⁰^,¹²

J. Cordero,

¹³

I. Harrison ,

¹³

C. Sanchez ,

²

J. Prat ,

¹⁴

D. Gruen ,

⁴^,⁵^,¹¹

H. Lin,

¹⁵

M. Crocce ,

¹⁶^,¹⁷

E. Rozo,

¹⁸

T. M. C. Abbott,

¹⁹

M. Aguena,

²⁰^,²¹

S. Allam,

¹⁵

J. Annis,

¹⁵

S. Avila,

²²

D. Bacon,

²³

E. Bertin,

²⁴^,²⁵

D. Brooks,

²⁶

D. L. Burke,

⁵^,¹¹

A. Carnero Rosell,

²⁷^,²⁸

M. Carrasco Kind,

²⁹^,³⁰

J. Carretero,

¹

F. J. Castander,

¹⁶^,¹⁷

A. Choi,

¹⁰

C. Conselice,

¹³^,³¹

M. Costanzi,

³²^,³³

M. Crocce ,

¹⁶^,¹⁷

L. N. da Costa,

²¹^,³⁴

M. E. S. Pereira,

³⁵

K. Dawson,

³⁶

S. Desai,

³⁷

H. T. Diehl,

¹⁵

K. Eckert,

²

T. F. Eifler,

³⁸^,³⁹

A. E. Evrard,

³⁵^,⁴⁰

I. Ferrero,

⁴¹

B. Flaugher,

¹⁵

P. Fosalba,

¹⁶^,¹⁷

J. Frieman,

¹⁵^,⁴²

J. Garc ´ıa-Bellido,

²²

E. Gaztanaga,

¹⁶^,¹⁷

T. Giannantonio,

⁴³^,⁴⁴

R. A. Gruendl,

²⁹^,³⁰

J. Gschwend,

²¹^,³⁴

S. R. Hinton,

⁴⁵

D. L. Hollowood,

⁹

K. Honscheid,

¹⁰^,¹²

B. Hoyle,

⁴⁶^,⁴⁷^,⁴⁸

D. Huterer,

³⁴

D. J. James,

⁴⁹

K. Kuehn,

⁵⁰^,⁵¹

N. Kuropatkin,

¹⁵

O. Lahav,

²⁶

M. Lima,

²⁰^,²¹

N. MacCrann,

⁵²

M. A. G. Maia,

²¹^,³⁴

M. March,

²

J. L. Marshall,

⁵³

P. Melchior,

⁵⁴

F. Menanteau,

²⁹^,³⁰

R. Miquel,

¹^,⁵⁵

J. J. Mohr,

⁴⁶^,⁴⁷

R. Morgan,

⁶

R. L. C. Ogando,

²¹^,³³

A. Palmese,

¹⁵^,⁴²

F. Paz-Chinch ´on,

³⁰^,⁴³

W. J. Perci v al,

⁵⁶^,⁵⁷^,⁵⁸

A. A. Plazas,

⁵³

M. Rodriguez-Monroy,

⁵⁹

A. Roodman,

⁵^,¹¹

G. Rossi,

⁶⁰

S. Samuroff,

⁶¹

E. Sanchez,

⁵⁹

V. Scarpine,

¹⁵

L. F. Secco,

²

S. Serrano,

¹⁶^,¹⁷

I. Sevilla-Noarbe,

⁵⁹

M. Smith,

⁶²

M. Soares-Santos,

³⁴

E. Suchyta,

⁶³

M. E. C. Swanson,

³⁰

G. Tarle,

³⁵

D. Thomas,

²³

C. To,

⁴^,⁵^,¹¹

T. N. Varga,

⁴⁷^,⁴⁸

J. Weller,

⁴⁷^,⁴⁸

and R. D. Wilkinson

⁶⁴

(DES Collaboration)

Affiliationsarelistedattheendofthepaper

Accepted2021November11.Received2021November11;inoriginalform2021January11

ABSTRACT

WepresentthecalibrationoftheDarkEnergySurveyYear3(DESY3)weaklensing(WL)sourcegalaxyredshiftdistributions n(z)fromclusteringmeasurements.Inparticular,wecross-correlatetheWLsourcegalaxiessamplewithredMaGiCgalaxies (luminousredgalaxieswithsecurephotometricredshifts)andaspectroscopicsamplefromBOSS/eBOSStoestimatetheredshift distributionoftheDES sourcessample.Twodistinctmethodsforusingthe clusteringstatisticsare described.Thefirstuses theclusteringinformationindependentlytoestimatethemeanredshiftofthesourcegalaxieswithinaredshiftwindow,asdone intheDES Y1analysis.The secondmethodestablishesalikelihood of theclusteringdataas afunctionofn(z),whichcan beincorporatedintoschemesforgeneratingsamplesofn(z)subjecttocombinedclusteringandphotometricconstraints.Both methodsincorporatemarginalizationovervariousastrophysicalsystematics,including magnification andredshift-dependent galaxy-matterbias.Wecharacterize theuncertaintiesof themethodsinsimulations;the firstmethodrecoversthemean zof tomographicbinstoRMS(precision)of∼0.014.Useofthesecondmethodisshowntovastlyimprovetheaccuracyoftheshape ofn(z)derivedfromphotometricdata.ThetwomethodsarethenappliedtotheDESY3data.

Keywords: galaxies:distancesandredshifts– cosmology:observations.

1 I N T R O D U C T I O N

TheDarkEnergySurvey(DES) isa photometricsurveythathas imaged5000deg²of the sky. TheDES Y3 ‘3x2’analysis (DES Collaboration2021)usingdatatakenduringthefirstthreeseasonsof

E-mail:[email protected](MG);[email protected](GG)

observationsconstrainscosmologicalparametersbycombiningthree differentmeasurementsoftwo-pointcorrelationfunctions:cosmic shear(Amonetal.2021;Seccoetal.2021),galaxy–galaxylensing (Pratetal.2020),andgalaxyclustering(Rodr´ıguez-Monroyetal.

2020).Thecosmicshearmeasurementprobestheangularcorrelation ofmorethan100000000galaxyshapesfromtheweaklensing(WL) sample(Gattietal.2021),dividedintofourtomographicbins.The cross-correlationofgalaxyshapesandthepositionsofredluminous

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galaxiesidentifiedbythe redMaGiCalgorithm(Rozoetal.2016) is measured by galaxy–galaxy lensing. Lastly, galaxy clustering measurestheautocorrelationofthepositionsofredMaGiCgalaxies.

A magnitude-limited sample (Porredon et al. 2021) will be also usedaslenssamplealternativelytoredMaGiCinasecondanalysis (Porredonetal.2021),withthegoalofimprovingthecosmological constraints.

The correct cosmologicalinterpretation of thesemeasurements relies on anaccurate estimateof the redshift distributionsof the samples,whichcanotherwiseleadtobiasesintheinferredcosmo- logicalparameters(e.g.Hutereretal.2006;Hildebrandtetal.2012; Choietal.2016;Hoyleetal.2018).Photometricsurveyshavebeen relyingon differentmethodologiesto deriveredshift distributions (Hildebrandtet al. 2010; S´anchez etal. 2014), mostly basedon galaxies’multibandphotometry(photo-zmethods,orPZ).However, these methodsare ultimatelylimited by the redshift ambiguities infew-bandcolours,andthelimitedandincompletespectroscopic samplesavailabletocalibratethecolour–redshiftrelations.

Clustering-zmethods(Newman2008;M´enardetal.2013;Choi et al. 2016; Davis et al. 2017; Johnson et al. 2017; Morrison etal.2017;Gattietal.2018;vandenBuschetal.2020)offeran alternativetostandardphoto-zmethodstoinferredshiftdistributions.

Inshort,clustering-zmethodsexploitthetwo-pointcorrelationsignal betweenaphotometric‘unknown’sampleanda‘reference’sample ofhigh-fidelityredshiftgalaxiesdividedintothinbins,toinferthe redshiftdistributionsofthephotometricsample.Oneofthebiggest advantagesofclustering-zmethodsisthatthereferencesampledoes nothavetoberepresentativeofthephotometricsample.Clustering-z methods(orWZ)havebeeninthepastyearssuccessfullyapplied to bothdata (Rahman etal. 2015, 2016a, b;Scottez etal.2016; Davis etal. 2017, 2018; Hildebrandt etal. 2017, 2021; Johnson etal.2017;Cawthonetal.2018;Batesetal.2019;vandenBusch etal.2020)andsimulations(McQuinn&White2013;Schmidtetal.

2013;Scottez et al. 2017; Gatti et al. 2018),and they represent onecrediblesupplementtostandardphoto-zmethodsforthenew, upcominggenerationofdatasets(Scottezetal.2017).

Clustering-zmethodshave beenused bothto provideaninde- pendentredshiftdistributionestimateandtocalibratedistributions inferredfromphoto-zmethods.IntheDESY1cosmologicalanal- ysis, weopted for the latterapproach (Daviset al. 2017;Hoyle etal.2018).Inparticular,weusedhigh-qualityphotometricredshifts providedbyredMaGiCgalaxies(Rozoetal.2016)tomeasurethe clustering-zsignalwiththeWLsource-galaxysample.Theuseof high-qualityphotometricredshiftsratherthanspectroscopicredshifts was motivated by the higher statistical power of the redMaGiC sample,owingtothelargenumberofredMaGiCgalaxies(650000 forDESY1)intheDESfootprint.Duetothelimitedredshiftrange oftheredMaGiCsample,clustering-zestimatescouldnothavebeen usedtodeterminen(z)initsentiretyontheirown,buttheyhavebeen usedtocalibratethemeanredshiftofthedistributionsmeasuredby otherDESphoto-zmethods(withthemeantakenovertheredMaGiC zbounds).AsimilarapproachhasbeenimplementedbytheKiDS team in theirrecentcosmological analysis (vanden Buschet al.

2020;Hildebrandtetal.2021),wherethey usedcross-correlation estimatestocalibratethemeanredshiftsinferredfromotherphoto-z methods.Theyusedanumberofdifferentspectroscopicsamplesas areferencesample,whichguaranteedagreaterredshiftcoveragebut lessstatisticalpowercomparedtotheuseofredMaGiCgalaxies.

ThestrategyforcalibrationoftheWLredshiftdistributionsfor DESY3improvesinmultiplerespectsontheY1strategyoutlinedin Gattietal.(2018).Fromtheclustering-redshiftside,weexecutetwo differentmethodsto combineclusteringinformationwithredshift

distributionsfromphotometry.Thefirstapproachistouseclustering- z to estimate the mean redshift zwz , and assign a clustering-z likelihoodtoanycandidaten(z)fromphoto-ztechniquesbasedon the valueofitsmeanzpz (similarto theDES Y1analysis). We willrefertothisasthe‘mean-matching’approach.Thesecond,new methodistoposeboththeclustering-zandthephoto-zmeasurements asprobabilitiesp[D|n(z)]oftheobservationaldataDgivenredshift distributions n(z); thento samplethe fulln(z)from the posterior p[n(z)]impliedbymultiplyingtheseprobabilities.Wewillreferto thisasthe‘full-shape’method.

WefurthermoreimproveoverY1inthemodellingoftheclustering signal,accountingfortheredshiftevolutionofthegalaxy-matterbias andtheclusteringoftheunderlyingdarkmatterdensityfield,which wereneglectedintheDESY1analysis.Inthesecondmethodthat calibratestheshapeoftheredshiftdistributions,wealsomarginalize over magnification effects.Finally, weusea combination oftwo differentreferencesamples: redMaGiCgalaxieswithhigh-quality photometric redshifts;and aspectroscopic samplefrom the com- binedBOSS(BaryonicOscillationSpectroscopicSurvey,Dawson etal.2013)andeBOSS(extended-BaryonOscillationSpectroscopic Survey,Dawsonetal.2016;Ahumadaetal.2020;Alametal.2021) catalogues.OnlyredMaGiCgalaxieswereusedinDESY1.Onone hand,redMaGiCgalaxiesspanthefullDESY3footprint(Rodr´ıguez- Monroyetal.2020)andarecharacterizedbyahighernumberdensity thanBOSS/eBOSSgalaxies,whichcoveronly≈17percentofthe DESY3footprint.Ontheotherhand,thelattersamplespansawider redshift rangeand hasbetterredshiftestimates, whichmakes the combinationofthetwosamplesdesirable.

Thefiducialphoto-zestimatesfortheDES Y3WLsampleare providedbyaself-organizingmap-basedscheme(hereafterSOMPZ, Buchsetal.2019;Mylesetal.2021).TheSOMPZmethodprovides ameans togeneratesamples ofthen(z) forall tomographicbins that encompass the uncertaintiesin the photometric inference of the distributions.Themean-matchingclustering-zmethod maybe usedtoconfirmoradjustthe n(z)samples generatedbySOMPZ.

Weusethefull-shapemethodasthefiducialmethodforDESY3, generatingsamplesofn(z)fromthecombinedSOMPZandclustering likelihoods.In eitherroute,the DESY3 cosmologicalanalysisis done by sampling over the finitesetof realizationsgenerated by SOMPZ+clustering-z.

Wenotethatthereexistotherstrategiestocombineclustering-z andphoto-zestimates.Forexample,S´anchez&Bernstein(2019)and Alarconetal.(2020)showhowtocombinephoto-zandclustering-z estimatesusingahierarchicalBayesianmodel(Leistedt,Mortlock

&Peiris2016).TheapplicationofthesemethodstoDESdataisleft forfuturework.

Thispaperisorganizedasfollows.InSection2,wedescribethe twodifferentmethodologiesusedinDES Y3to calibratephoto-z posteriorsusingclustering-zestimation,andexplainhowtoassign a likelihoodto the cross-correlation information.Thesimulations andthedatasetsusedinthispaperaredescribedandcomparedin Section 3.InSection4,weperformextendedtestsinsimulations assessingthesystematicuncertaintyofthemethods.Thecalibration onDESY3dataispresentedinSection5,andinSection6wediscuss futureprospectsforthismethodandpresentourconclusions.

2 M E T H O D O L O G Y

We describe the clustering-z (WZ) methodology as generally as possibleinthissection,deferringtoSection3thedescription(and thechoiceofthebinning)oftheparticularsamplesadoptedforDES Y3.

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2.1 Modellingandmeasuringthecorrelationsignal

Clustering-z methods rely on the assumption that the cross- correlationbetweentwosamplesofobjectsisnon-zeroonlyinthe caseofoverlapofthedistributionofobjectsinphysicalspace,due theirmutualgravitationalinfluence.Letusconsidertwosamples:

(i)Anunknownsample,whoseredshiftdistributionn_u(z)hasto bemeasured,namelyourWLsourcesample,and

(ii) Areferencesample,whoseredshiftdistributionn_r(z)isknown (eitherfromspectroscopicredshiftsorfromhigh-precisionphoto- metricredshifts).

Wecomputetheangularclusteringsignalw_urasafunctionofthe separationangleθbetweentheunknownsampleandthereference population.UndertheassumptionoflinearbiasingandtheLimber approximation(Limber1953),theclusteringsignalcanbewrittenas (e.g.Krauseetal.2017):

w_ur(θ)=

dzn_u(z)n_r(z)b_u(z)b_r(z)w_DM(θ,z)+M(θ), (1)

where n_u(z) and n_r(z) are the unknown- and reference-sample redshift distributions (normalized to unity over the full redshift interval),bu(z)andbr(z)arethelineargalaxy-matterbiasesofthe twosamples,and w_DM(θ,z)isthedark-mattertwo-pointangular correlation function. Theterm M(θ) refersto the contribution of lensingmagnificationeffects; descriptionandfullexpressionsfor thetermswDM(θ,z)andM(θ)aredetailedbelow(equation7and equationA1).Notethatwhileweacknowledgethattheassumption of linearbiasing is not expectedto holdat small scales, we are nevertheless confident to be able to estimatethe systematicbias introducedbythispremise,asexplainedinSection2.2.Wealsonote thattheLimberapproximationisastandardassumptioninclustering- zworks,anditisexpectedtohaveaminimalimpactonourresults (e.g.McQuinn&White2013).

FollowingM´enardetal.(2013),thecorrelationfunctionismea- suredas afunctionofangle, andaveragedover angularscales to producea‘scalar’valuevia

¯ w_ur=

θ_max θmin

dθ W(θ)w_ur(θ), (2)

where W(θ)∝θ⁻^γ is a weighting function. We adopt γ = 1 to yieldoptimalS/Nonthescalarin thepresenceofshotnoise.The integrationlimitsintheintegralinequation(2)correspondtofixed physicalscales.Inthiswork,wechoosetospanthephysicalinterval between1.5and5.0Mpc(Section4).WeusetheDavis&Peebles (1983)estimatorforthecross-correlationsignal,

w_ur(θ)= NRr

NDr

DuDr(θ)

DuRr(θ) −1, (3)

whereD_uD_r(θ)andD_uR_r(θ)are,respectively,data–dataanddata–

randompairs.Thepairsare properlynormalizedthroughN_Drand N_Rr,correspondingtothetotalnumberofgalaxiesinthereference sampleand inthe referencerandom catalogue.Ifweightsforthe referencecatalogueofgalaxies(orforthecatalogueofrandoms)are provided,NDr(orNRr)isthesumoftheweightsofthecatalogue, andD_uD_r(θ)(orD_uR_r(θ))istheweightednumberofpairs.Notethat weightscanalsobeassignedtotheunknownsample;inthatcase, theweightednumberofpairs D_uD_r(θ)(orD_uR_r(θ))alsoaccounts forthe weightsofthe unknownsample. Asin Gattietal.(2018), we use the Davis & Peebles estimator rather than the Landy &

Szalay(1993)estimatorsincetheformerinvolvesusingacatalogue ofrandompointsforjustone ofthe twosamples.Thisallowsus

to avoidcreatinghigh-fidelityrandomcataloguesfortheDESY3 sourcegalaxysample,whoseselectionfunctionisverycomplexand non-trivial toreplicate,besidesbeingcomputationallyverycostly.

For ouranalysis,weonlyrelyon randompointsforthe reference sample,whoseselectionfunctionandmaskarewellunderstood.We notethatintherestofthepaperweadoptedtheDavis& Peebles estimatorevenwhenmeasuringtheautocorrelationofthereference samples,butwecheckedthatusingtheLandy&Szalayestimator leadtonegligiblevariations.

Nowweassumethatthereferencesampleisdividedintoredshift bins centredatz_i, eachnarrowenough that wecanapproximate nr,i(z)≈δD(z−zi),withδDbeingDirac’sdeltadistributionandthe integrandsinequation(1)otherthann_rcanbetreatedasconstant.

Equations(1)and(2)become:

¯

w_ur(z_i)≈n_u(z_i)b_u(z_i)b_r(z_i)w¯_DM(z_i)+M¯(z_i), (4) where barred quantities indicate they have been averaged over angular scales as per equation (2). In what follows we will, for simplicity,dropthebar.Theabovequantityisalwaysestimatedat theredshiftziofthei-ththinreferencesamplebin.

Thegoalistouseequation(4)toinfern_u(z),theunknownredshift distribution,fromthemultiplemeasureswur(zi).Butitisimportant to notethat thisequation followsfroma simplifyingassumption.

We assumed the galaxy-matter bias to be described by a single numberatallscales;thisistrueatlargescalesinthelinearregime, butwedo notexpectthis toholdatthe smallscalesusedinthis work (1.5 to 5.0 Mpc). In the non-linear regime, even the fact thatthetermsinsidetheintegralfactorizesintob_r(z_i)b_u(z_i)w_DM(z_i) is not guaranteed(Bernardeau et al. 2002;Desjacques, Jeong &

Schmidt2018).Thelinear-biasassumptionintroducesasystematic uncertaintythatdependsonthescalesadoptedandthesamplesunder studyandthatwillbequantifiedinthefollowingsections.

Theevolutionofthequantitiesb_r(z_i),b_u(z_i),w_DM(z_i)andM(z_i) needstobecharacterizedtocorrectlyrecovertheredshiftdistribution oftheunknownsample.Weturnnowtohowtomodelorestimate theseterms.

(i)Thegalaxy-matterbiasevolutionofthereferencesample b_r(z).Aslongastheredshiftsofthereferencesampleareaccurate enough, and we assume linear biasing, we can estimate b_r(z) by measuring the angle-averaged estimate of the autocorrelation functionofthereferencesampledividedintothinredshiftbins(δz

=0.02)centredatzi: w_rr(z_i)=

dz

b_r(z)n_r,_i(z)2

w_DM(z). (5)

Ifthebinsaresufficientlynarrowsoastoconsiderthebiasesand w_DMconstantoverthedistributions,theycanbepulledoutofthe aboveintegrals:

w_rr(z_i)=b_r²(z_i)w_DM(z_i)

dzn²_r,_i(z). (6)

Knowledgeoftheredshiftdistributionsofthenarrowbins isthen requiredto useequation(6)toestimateb_r(z_i).Lastly,weneedto modelw_DM(z)tocorrectlyrecoverb_r(z).

(ii)Thegalaxy-matterbiasevolutionoftheunknownsample b_u(z). In principle, the autocorrelation of the unknown sample constrainsthis.Howeverinourcase,nu(z)isbroadandunknown,and b_ulikelyvariessubstantiallyacrossthesample,sotheinformationon b_ufromtheautocorrelationisweakandentangledwithn_uitself.The degeneracybetweenbu and nu is the fundamentallimitingfactor of clustering-z methods. Mitigation schemes exist, basedon the useofadditionalinformationtoconstraintheevolutionofbu:e.g.

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Matthews&Newman(2010)usetheadditionalconstraintscoming fromtheautocorrelationfunctionofthetomographicbins(without dividing the samples intothin bins);or the method implemented invanden Buschetal.(2020),whousethe additionalconstraint coming fromthe normalizationof the redshiftdistribution ofthe fullunknowncataloguenotdividedintotomographicbins.However, thesemethodsarenotfreefromshortcomings,sowedecidednot to attemptcorrectingfor b_u.Since it isdifficult to place apriori constraints on b_u, when forward modelling the clustering signal wechoseto parametrize it in aflexible way (seeSection 2.2.2), effectivelytreatingitasafreefunction.

(iii)Thedarkmattertwo-pointcorrelationfunctionwDM(z). Thiscanbemodelledassumingagivencosmologyandanon-linear powerspectrum.Atfixedz_i,thiscanbewrittenas:

wDM(zi)=

dθ W(θ)2+1 4π P(cosθ)

× 1

χ(zi)²H(zi)P_NL

l+1/2 χ(zi) ,z_i

, (7)

whereχisthecomovingdistanceandH(zi)istheHubbleexpansion rateatredshiftz_i.P(x)istheLegendrepolynomialoforder.P_NL(k, χ) isthe3D non-linear matterpower spectrumatwavenumberk (which,intheLimberapproximation,issetequalto(l+1/2)/χ(z_i)) andatthecosmictimeassociatedwithredshiftzi.Wefindthatthe redshiftevolutionofw_DM(z_i)dependslittleontheparticularvalue ofcosmologicalparameters,whereasthedependenceoftheoverall amplitudeofwDM(zi)withrespecttocosmologyisabsorbedbyour systematicfunctions.Basedonthis,weholdcosmologyfixedwhen computingwDM(zi), assumingthe valuesin PlanckCollaboration VI(2020).Wethen verifyaposteriorithat thisapproximation is validby repeatingouranalysisusingverydifferentvaluesforthe cosmologicalparameters ( _m= 0.4, σ₈= 0.7), findingthat the impact on our conclusions is negligible. Note that some of the mitigationschemesadoptedinliteraturetocorrectthegalaxy-matter biasevolutionoftheunknownsamplealsoautomaticallyestimate wDM(zi)fromthedata(Matthews&Newman2010;vandenBusch etal.2020),buttheyarenotadoptedinthiswork.

(iv)Magnification signal M(z_i). WL magnification (Narayan 1989;Villumsen,Freudling&daCosta1997;Moessner&Jain1998) changestheobservedspatialdensityofgalaxies:theenhancement inthe fluxofmagnifiedgalaxiescanlocallyincreasethe number density,asmoregalaxiespasstheselectioncuts/detectionthreshold ofthesample;atthesametime,thesamevolumeofspaceappears to cover adifferentsolid angleon the sky, generallycausing the observednumberdensitytodecrease.Foraflux-limitedsample,the neteffectisdrivenby theslopeof theluminosityfunctionofthe sample, hereconvenientlyparametrized throughthe parameterα, andithasanimpactonthemeasuredclusteringsignal.Formally,the magnificationtermdependsonthegalaxy-matterbiasandparameter α of the two samples, as well as on the redshift distribution of theunknownsample:M(z_i;α_r,α_u,b_r,b_u,n_u).Moredetailsabout ourmodellingofthemagnificationeffectsaregiveninAppendixA, althoughweanticipatemagnificationeffectshaveanegligibleimpact onouranalysis,duetoouranalysischoices.Tokeepournotation light,whenpossible,wewillsimplyindicatemagnificationeffects asM(z_i),droppingthedependenceonotherfactors.

Undertheassumptionofthinreferencebins,lineargalaxy-matter bias, and using the linearized version of the equation describing magnificationeffects(AppendixA),equation(4)becomesalinear systemofequations,andcanbesolvedtoobtainanestimateofn_u(z_i).

Thiswouldbesimilartostandardclustering-zmethodswhichuse

thecross-correlation signalasastartingpointtoinfertheredshift distributionsoftheunknownsample(Newman2008;McQuinn&

White2013;M´enardetal.2013;Schmidtetal.2013).

Alternatively,if anestimateof then_u(z_i)is providedby e.g.a photo-zmethod,equation(4)canbeusedtoevaluatetheexpected correlationsignalw_ur(z_i)andcompareittotheonemeasuredindata, i.e.aforwardmodellingapproach(seee.g.Choietal.2016).

Thisworkrepresentsa significantadvancementover DES Y1, becauseintheY1analysisnoneofthetermsdescribedabovewere modelled. Weassumed br(zi),bu(zi), and wDM(zi) to beconstant within eachphoto-z bin,and usedthe simulationstoestimatethe systematicerrorinducedbythisassumption.InDESY1wealsodid notmodelM(z_i),butwedecidedtoexcludetheredshiftrange(i.e.

the tails of the redshiftdistributions) wheremagnification effects areexpectedtohaveanon-negligibleimpact.Onthecontrary,inthis workwemodelb_r(z_i),w_DM(z_i),and,dependingonthemethod,M(z_i).

2.2 Assigninglikelihoodtothecross-correlationinformation Weusetheclusteringdata{w_ur(z_i),w_rr(z_i)},toplacealikelihood L[WZ|n_u(z)]ofobtainingtheclustering-zdatagivensomeestimate ofthetruenu(z).Theclustering-zdatawillbeusedtoevaluatethe likelihoodofmanycandidaten_u(z)functions,typicallydrawnfrom some combination of PZand spectroscopicdata.In the DES Y1 analysis,suchrealizationsweretakenasn_u(z)=n_pz(z+z),where npz(z)wasasingle‘best’photo-zestimateandzafreeparameter.

The Y3 approach ismoregeneral, withmany realizationsof the fullfunctionn_u(z)beingdrawn.Inanycaseweneedonlytodefine L[WZ|n_u(z)].Todoso,wemakeuseoftwoapproaches,described below.

2.2.1 Mean-matchingmethod

Thismethodworksby compressingthen(z) functionstoasingle statistic,theirmeanz.Inthis‘simpler’method,wedonotmodel magnificationeffects,sothemeanistakenoverarestrictedrangeof z,whereareferencesampleisavailableandw_ur(z)M(z),such thatwecanneglectmagnificationeffects.Forthismethod,cuttingthe tailscanbepreferableevenwhenestimatesofmagnificationeffects inthetailsareavailable.Thisisduetothefactthatsmallerrorsin themagnificationestimatesinthetailscanhavealargeimpacton themeanoftheredshiftdistribution,loweringthecapabilityofthe methodtoconstrainthemeanredshift.

FollowingtheDESY1analysis,wechooseafixedinterval[z_min, z_max] = [zpz − 2σ_pz, zpz + 2σ_pz], where zpz and σ_pzare the meanandrootmeansquareofacanonicaln_pz(z).Incasethe fixedintervalincludesarangewherethereisno referencesample coverage,itisfurtherreducedtoensurethereareenoughgalaxiesin thereferencesampletoprovideameaningfulclustering-zestimate (seeSection4.1formoredetails).Wefirstcreateanominal‘naive’

estimatorn˜_u(z)usingequation(4)whichwouldbeproportionalto anunbiasedestimatoriflinearbiasholdsandbu(z)isconstant:

˜

nu(zi)∝ w_ur(z_i)

b_r(z_i)w_DM(z_i), (8) Then wedefine mean redshifts forthe clustering-z data and the proposedn_pz(z)as

zwz= _z_max

z min dzzn˜_u(z) z_max

z min dzn˜_u(z) (9)

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zpz = z_max

z min dzzn_pz(z) _z_max

z min dzn_pz(z) (10)

ThelikelihoodoftheWZdatagivenaproposedn_u(z)isthentaken tobeaGaussiandistributioninthezvalues:

L[WZ|n_u(z)]≡N zpz − zwz ,σ_z

(11) Theuncertaintyσ z mustincorporatetheestimatedmeasurement noiseandalsosystematicerrorsfromshortcomingsoftheunderlying model.Section4.1givestheresultsofusingsimulationstosetthese uncertainties.TheassumptionofGaussianityisareasonablechoice even in absence of systematics, as per the central limit theorem (themeanredshiftcompressestheinformationfrommanydifferent redshifts). Moreover, we parametrize the impact of systematics effectsinsuchawaytheycanbedescribedbyaGaussianlikelihood, andsystematiceffectsdominateourtotalerrorbudget.

2.2.2 Full-shapemethod

This method dispenses with the mean statistic and simply compares the observed w_ur(z_i) data to a model

ˆ

wur[zi;nu(z),br(z),bu(z),αr(z),αu(z),s]thatincorporates potential systematiceffects.Themodelisanalterationofequation(4):

ˆ

wur(zi)=nu(zi)br(zi)wDM(zi)×Sys(zi,s)+M(zi). (12) The functions n_u(z), b_r(z), and M(z) are assumed to be given beforehand,andwDMiscalculatedfromtheoryasdescribedinequa- tion(7).TheSysfunctionmultipliestheclusteringsignalbysome redshift-dependentvalue thatis parameterized by s={s₁,s₂,...} that wewillmarginalize over. Theroleof the Sys functionis to absorballuncertaintiesinb_uanditsredshiftdependence,aswellas uncertaintiesduetofailuresinthelinearbiasmodelitself,andinthe determinationofb_r(z).ThechoiceofSysfunctionandthepriorson itsparametersareguidedbysimulationsasdescribedinSection4.2.

Asaruleofthumb,weexpecttheSysfunctionamplitudetoslowly varyacrossredshift,andtobeofthe samemagnitudeofatypical galaxy-matterbias(i.e.aroundunity).Wenotethatinprinciplewe couldalsohaveabsorbedthe redshiftdependence ofw_DM,or the magnificationcontributionM(z),intotheSysfunction.Wedidnot proceedthiswaysinceweknowhowtomodelthesecontributions, althoughthiscomesattheexpenseofamorecomplexmodel.Lastly, wenotethatformallythemagnificationcontributionalsodepends on the bias bu; this ismarginalized separately, together with the magnificationparameteroftheunknownsampleα_u(moredetailsare giveninAppendixA).

Withamodelforw_urinhand,weassumethatthemeasurement errorsinthedataareGaussiananddefinealikelihood

L[WZ|nu(z),br(z),αr(z),wDM(z)]

∝

dsdpexp

−1

2(w_ur−wˆ_ur)^T_w⁻¹(w_ur−wˆ_ur)

p(s)p(p), (13) wherep={bu,αu}entersinthemodellingofthemagnificationterm.

Thedata andmodelforw_urare takenhereto bevectors overz_i, andw isthecovariancematrixofthe data(fromshotnoise and samplevariance).Thenuisanceparametersetssand peachhave theirownpriors.Itisthe extentof thesepriors thatregulates the levelofsystematicerrorallowedforintheinferenceofnu(z)from theclustering-zdata.Thesystematicfunctionandthesepriorsare quantifiedinSection4.2.

The covariance matrix _wis estimated from simulated data throughajackknife(JK)approach,usingthefollowingexpression (Quenouille1949;Norbergetal.2009):

ˆ(xi,xj)= (N_JK−1) N_JK

N JK

k= 1

x_i^k−x¯i

x_j^k−x¯j

, (14)

wherethesampleisdividedintoNJK=1000subregionsofroughly equalarea,x_iisameasureofthestatisticofinterest(=w_ur)inthe i-thbinofthek-thsample,andx¯i isthemeanoftheresamplings.

The jackknife regions are safely larger than the maximum scale consideredinourclusteringanalysis.ThecorrectionfromPercival etal.(2021)isimplementedwhencomputingtheinversecovariance, althoughithasamodestimpact(∼10percentontheamplitudeof thecovariance)giventhenumberofjackkniferegionsandthedata vectorlength.

Notethattheclustering-zlikelihoodinequation(13)dependsex- plicitlyontheestimatedbiasandmagnificationcoefficientb_randα_r ofthereferencesample,anddependsimplicitlyonthecosmological modelthroughthedark-matterclusteringw_DM.Thusinprinciple,this likelihoodandtheinferencesonnu(z)mustberecalculatedforeach changeincosmologicalmodel.Wehave,however,testednumerically thatthefullexpressionforL[WZ|n(z)]hasnegligibledependenceon thecosmologicalparametersorthereference-samplepropertiesonce themarginalizationoversystematicnuisancessandparedone.This isbecausethesystematicvariableshaveenoughfreedomtoabsorb thesmallchangesinthemodelwroughtbychangesincosmology.

It isthereforeallowable forusto compute equation(13) using a fiducial cosmology and fiducial values ofb_rand α_r, and usethe inferredredshiftdistributionsinacosmologicalinferencethatmight varytheseparameters.

3 DATA A N D S I M U L AT E D DATA

This section describesthe variousphotometric and spectroscopic catalogues thatfeed intothe clustering-z measurements.The full analysisisalsoconductedonsimulatedcatalogues;foreachelement oftherealanalysis,wealsodescribehowitssimulatedcounterpart wasgenerated.

3.1 DESY3data

TheDESobserved∼5000squaredegreesoftheSouthernhemisphere in five differentbroadphotometricbands (grizY) over 6yrusing the Dark EnergyCamera(DECam,Flaugher etal.2015), a570- megapixelcamerabuiltbytheDESCollaborationandstationedat the CerroTololoInter-AmericanObservatory(CTIO)4-mBlanco telescope.DESwillmeasuretheshapesofabout300milliongalaxies uptoredshiftz∼1.4.Inthispaper,wefocusontheanalysisofthe first3yr(Y3) ofobservations.DESY3dataspanthefullareaof thesurvey,4143deg²aftermaskingforforegroundsandproblematic regions,amajoradvanceoverthe 1321deg²ofDES Y1(Drlica- Wagner etal.2018;Troxeletal.2018).Thecomplete DES(Y6) reachesgreaterdepththanY3data;furthermore,thedataaremore uniformindepth..ThetotalnumberofobjectsdetectedinDESY3 is≈390000000.Objectdetectionandmeasurementsaredescribed inSevilla-Noarbeetal.(2021).

3.2 BuzzardN-bodysimulation

We use one realization of the DES Y3 Buzzard catalogue v2.0 (DeRoseetal.2019).Initialconditionsweregeneratedusing2LPTIC

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Figure1. RedshiftdistributionsoftheredMaGiCsamples,binnedusingthe redMaGiCphoto-zestimates,indataandinsimulations.

(Crocce,Pueblas& Scoccimarro2006)andthe N-bodyrunusing L-GADGET2(Springel2005).Cosmologicalparametershavebeen chosentobe m=0.286,σ8=0.82, b=0.047,ns=0.96,h=0.7.

Light-conesaregeneratedontheflystartingfromthreeboxeswith differentresolutions andsize (1050³, 2600³, and 4000³ Mpc³h⁻³ boxesand1400³,2048³,and2048³particles),toaccommodatethe needofalargerboxathighredshift.Haloesareidentifiedusingthe publiccodeROCKSTAR(Behroozi,Wechsler&Wu2013)andthey arepopulatedwithgalaxiesusingADDGALS (DeRoseetal.2019).

Galaxiesareassignedmagnitudesandpositionsbasedontherelation betweenredshift,r-bandabsolutemagnitude,andlarge-scaledensity foundinasubhaloabundancematchingmodel(Conroy,Wechsler&

Kravtsov2006;Lehmannetal.2017)inhigherresolutionN-body simulations. SEDsare assigned to galaxiesfrom the SDSS DR7 ValueAddedgalaxycatalog(Blantonetal.2005)byimposingthe matchingwiththe SED–luminosity–densityrelationshipmeasured intheSDSSdata.SEDsareK-correctedandintegratedovertheDES filterbandstogenerateDESgrizYmagnitudes.Lensingeffectsare calculatedusingthemultipleplaneray-tracingalgorithmCACLENS

(Becker2013),whichprovidesweak-lensingshear,magnification, andlensedgalaxypositionsforthelight-coneoutputs.

3.3 Referencesample1:redMaGiCgalaxies

Thefirstreferencesampleusedinthisclustering-zanalysisconsists ofDESredMaGiCgalaxies.TheredMaGiCalgorithmselectsred luminousgalaxieswithhigh-qualityphotometricredshiftestimates (Rozo etal. 2016). This is achieved by fitting each galaxyto a redsequencetemplate;galaxiesarethenselectedonlyiftheypassa goodnessoffitandluminositythreshold.InDES,redMaGiCgalaxies areusedaslenssampleinthegalaxy–galaxylensinganalysisand intheclusteringanalysis(Pratetal.2020;Rodr´ıguez-Monroyetal.

2020).Twosamplesareselectedwithdifferentnumberdensityby meansof twodistinctluminositythresholds:afirstsample called

‘highdensity’selectedwithacutL/L∗>0.5andasamplecalled

‘highluminosity’selectedwithacutL/L∗>1.Acombinedsample isthenobtainedbyjoiningthesetwosamples,usingthehigh-density sampleforredshiftsz<0.65,thehigh-luminositysampleforhigher redshifts.

Insimulations,theredMaGiCsampleisselectedwiththesame algorithm used in the data. A comparison between the redshift distributionsfortheredMaGiCsamplesindataandinsimulations isshowninFig.1,illustratingthegoodagreementbetweenthetwo.

Small differences are due to smalldiscrepancies in the evolution ofthe red-sequencebetweenthesimulationandthe data.Bothin simulationsand in data,theredMaGiCsampleis dividedinto40 bins of width z = 0.02 spanning the 0.14 < z < 0.94 range of the redMaGiC catalogue.¹ The particular choice of the bin widthis notexpected toimpactour conclusions,as longas bins are small enoughcompared to the typicalvariation scales of the WL n(z) and the galaxy-matter biases of the two samples. The total number ofredMaGiC galaxies is3041935 in the data, and 2594036 inthe simulation.The differencein thenumberdensity is due to the aforementioned discrepancy in the evolution of the red-sequencebetweendata and simulations.This impliesthat the statisticaluncertaintiesoftheclustering-zestimatesobtainedusing theredMaGiCsamplearelargerinsimulationscomparedtodata.We donotexpectthistobeimportant,asweshowinSection4.1thatthe clustering-zmethodologyisdominatedbysystematicuncertainties, andthestatisticaluncertaintiesarenegligible.

WecomparethetypicalredMaGiCphoto-zscatterandbiasfound indataversusinsimulationsinFig.2.Sinceonlyaportionofthe datahavespec-zinformation,wereweightthemagnitudedistribution of the spectroscopic sample such that it matches the magnitude distributionoftheredMaGiCgalaxiesbeforecomputingthestatistics showninFig.2.Thisreweightingisperformedseparatelyforeach redshiftbin.Notethatthe typicalscatterofredMaGiCphoto-zis similartoourbinwidth,whichmightcallintoquestionthechoiceof binwidthforredMaGiCgalaxies.However,weverifyinSection4.1 thatevenwiththisset-up,redMaGiCphoto-zuncertaintiesarenota dominantsourceofsystematicerrorforourmethodology.Therefore, wedecidedthatusingalargerbinwidthforredMaGiCgalaxieswas notnecessary.

Usingcross-correlationtechniques,Cawthonetal.(2020)noted thatphoto-zuncertaintiesinredMaGiCgalaxiesatz>0.8might be underestimated. We do not think this constitutes a problem for the current analysis, as redMaGiC photo-z uncertainties are a subdominant systematicin our methodology(Section 4.1),and clustering-zconstraintsatz>0.8aredrivenbytheBOSS/eBOSS sample(Section4.2.2).

AcatalogueofrandompointsforredMaGiCgalaxiesisgenerated uniformlyoverthefootprint.Bothindataandinsimulations,weights areassignedtoredMaGiCgalaxiessuchthatspuriouscorrelations with observational systematics are cancelled. Note that due to low-statistics issues, the weights do not resolve fluctuations on scalesrelevantforthiswork,butonlycapturelarge-scalespurious correlations.Themethodologyusedtoassignweightsisdescribed in Rodr´ıguez-Monroy et al. (2020), and it is the same for data and simulations. The maindifference between data and Buzzard simulationsisthatthelatteronlymodelsdepthvariationsacrossthe footprint,whiledataaresubjectto alargernumberofsystematics whicharenotmodelledinsimulations.Thisshouldnotaffectany conclusiondrawnhere:theweightsremovethespuriousdependence ofthenumberdensitywithrespecttoanysystematic,regardlessof theirnumber,atleastatthelevelneededfortwo-pointcorrelation functionstobeunbiased(Rodr´ıguez-Monroyetal.2020).Thisof course holdsas longas all the systematicsaffecting the data are takenintoaccountwhenproducingtheweights.

1WenotethatthesimulatedredMaGiCsamplespansaslightlywiderrangein redshift;wenonethelesscuttheredshiftintervalatz=0.90tobeconsistent withthedata.

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Figure2. Thebias(left)andscatter(right)ofzredMaGiCforthesimulatedredMaGiCsample(solidlines)comparedtothedata(dashedlines).

Figure3. Spatialcoverageofthetworeferencesamplesusedinthiswork.

Purple indicates thecoverageby redMaGiC galaxies, pinkindicates the coveragebyBOSSandeBOSSgalaxies.

3.4 Referencesample2:spectroscopicgalaxies

Thesecondreferencesampleusedinthisworkisacombinationof spectroscopicsamplesfromthe SloanDigital SkySurvey(SDSS, Gunn et al. 2006; Eisenstein et al. 2011; Blanton et al. 2017).

In particular, we combine SDSS galaxies from BOSS (Dawson et al. 2013;Smee et al. 2013) and from eBOSS(Dawson et al.

2016;Ahumadaetal.2020;Alametal.2021).TheBOSSsample includestheLOWZandCMASScataloguesfromtheSDSSDR12, fullydescribedinReidetal.(2016),whileweincludedthelarge- scalestructurecataloguesfromemission-linegalaxies(ELGs, see Raichooretal.2017forthetargetselectiondescription),luminous redgalaxies(LRGs,targetselectiondescribedinPrakashetal.2016), and quasi-stellarobjects (QSOs)(Hou etal.2021) from eBOSS, whichwereprovidedtoDESforclustering-zsusagebyagreement betweenDESandeBOSS.Thedifferentsamplesarestackedtogether, andusedasonesinglereferencesampleinthiswork.Eachsample comeswithitsowncatalogueofrandompoints,whichaccountfor selectioneffects.Differentcataloguesofrandompointsarestacked together.Wemadesure the ratioofthe numberof randomswith respectto the numberofgalaxies wasthesameforeachrandom cataloguebeforecombiningthem.Bothinsimulationsandindata, theBOSS/eBOSSsampleisdividedinto50binsspanningthe0.1<

z<1.1rangeofthecatalogue(widthz∼0.02).Theareacoverage issmallercomparedtoredMaGiC galaxies,asshowninFig. 3.The

Table1. ListofthespectroscopicsamplesfromBOSS/eBOSSoverlapping withtheDESY3footprintusedasreferencegalaxiesforclustering-zsinthis work.

Spectroscopicsamples

Name Redshifts Ngal Area

LOWZ(BOSS) z∼[0.0,0.5] 45671 ∼860deg² CMASS(BOSS) z∼[0.35,0.8] 74186 ∼860deg² LRG(eBOSS) z∈[0.6,1.0] 24404 ∼700deg² ELG(eBOSS) z∈[0.6,1.1] 89967 ∼620deg²

QSO(eBOSS) z∈[0.8,1.1] 7759 ∼700deg²

redshiftdistributionofthesamplesisshowninFig.4,andthearea coverageandnumberofobjectsofeachsamplearesummarizedin Table1.NotethatsomeofthegalaxiesintheBOSS/eBOSSsample arealsointheredMaGiCcatalogue:∼1percentoftheredMaGiC galaxiesarematchedto∼10percentoftheBOSS/eBOSSgalaxies, within1arcsec.WedidnotremovethesegalaxiesfromtheredMaGiC sample,astheyhaveanegligibleimpactbothonourconstraintsand onthecovariancebetweenthetwosamples(asitwillbeclearinthe followingsections,theconstraintsfrombothsamplesaresystematic- dominated).

Toreplicatethe spectroscopicBOSS/eBOSSsamplein simulations, weselected bright galaxies withsimilar sky coverage and redshift distributionasthe onesindata.Wedidnot trytofurther match otherpropertiesof the sample, e.g. the galaxy-matterbias likelydiffersfromthatoftherealdata.Wenotethattheclustering-z methodologycorrectsforthereferencebias,soatnopointinthe analysisoftherealdataareweassumingthatthesimulationshave thesamebias.

3.5 WLsample

The WL sample in data is created using the METACALIBRATION

pipeline,whichisfullydescribedinGattietal.(2021).Aftercreation of theDESY3‘Gold’catalogue(Sevilla-Noarbeetal.2021),the

METACALIBRATION pipelinemeasures the shapesof eachdetected object. Selectioncutsfor thesample are describedin Gattietal.

(2021)andarechosenfromresultsoftestsonbothskydataandimage simulations(MacCrannetal.2022),andaredesignedtominimize systematicbiasesintheshearmeasurement.Galaxiesareweighted by the inverse variance of shear measurement, which increases the statisticalpowerofthecatalogue.Thefinalsamplecomprises 100204026objects,foraneffectivenumberdensityofn_eff=5.59 galarcmin⁻². Galaxies are furtherdivided into four tomographic

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bins,andredshiftdistributionestimatesforeachofthetomographic binsareprovidedbytheSOMPZmethod(Buchsetal.2019;Myles etal.2021).Thetomographicbinsareselectedsuchthattheyhave roughlyequalrawnumberdensity.

TheWLsampleisreproducedwithhighfidelityintheBuzzard simulationbyapplyingfluxandsizecutstothesimulatedgalaxies thatmimictheDESY3sourceselectionthresholds.TheWLgalaxy sampleinBuzzardisselectedwiththeaimofreproducingthesame selectionappliedinDESY3dataintermsofsize,signal-to-noise ratio,andcolours. Shapenoisehasbeen addedto thegalaxiesto matchthemeasuredshapenoiseoftheDESY3WLsample.

3.5.1 Photo-zestimates:SOMPZ

TheSOMPZmethodusesspectroscopicandmultibandphotometric information,anddatafromanumberofdeepfields(Hartleyetal.

2022)whereadditionalphotometryintheinfraredbandsYJKsand u-bandisavailable,besidesthestandard5-band(grizY)photometry availableintheDESwidefield.Thisadditionalinformationisused tobreakthedegeneraciesinthephoto-zestimatesoftheDESwide- fieldgalaxies(whichhavefewerbandsavailable).Thisisachieved bycreatingtwoSelf-OrganizingMaps(SOM,Kohonen1982),one mappingthedeep/spectroscopicgalaxiesintoa2Dgridofcellsusing their8-bandfluxes,andanothermappingthe WLsamplegalaxies intoa2D gridusing theriz photometry.Aprobabilistic mapping fromthewide-fieldSOMtothedeep-fieldSOMisgeneratedusing the‘Balrog’source-injectionsimulations(Everettetal.2020)and amapfromthe deep-fieldSOMtoredshiftisestimatedusingthe spectroscopicdata.

Thetomographicbinsareconstructedasfollows:afirstsetofedge valuesarearbitrarilyselected.Eachgalaxyoftheredshiftsampleis thenassignedtothetomographicbininwhichitsredshiftestimate falls.Anumberofgalaxiesatthispointsharethesamephotometry cellofthewide-fieldSOMandsametomographicbin,sothecellin itsentiretyisassignedtothebintowhichthemajorityofitsgalaxies live.Theinitialbinedgesare adjustedtoyieldapproximatelythe samenumberofgalaxies,andfinallythewholeprocedureisrepeated withthenewbinedges.Aftercompletingthisprocedure,thefinal binedgesare[0.0,0.358,0.631,0.872,2.0]fortheY3WLsource catalogue.

The full Y3 SOMPZ procedure is described in Myles et al.

(2021).Anumberoffactors contributeto theerror budgetofthe method:(1)shotnoise(i.e.thelimitednumberofgalaxyredshifts available);(2)samplevariance(i.e.thefactthatthespectroscopicand deepfieldsspanalimitedarea);(3)systematicuncertaintiesinthe spectroscopic/multibandphotometrysamples;(4)uncertaintyinthe methodologyingeneral;(5)photometriccalibrationuncertaintiesin theY3deepfields,i.e.theuncertaintyonthezero-pointcalibration ineachband.

Thetotalerrorbudgetisdominatedbythephotometriccalibration uncertaintyinthelow-redshiftbin,whileitisdominatedbysample variance and biases in the spectroscopic/multiband photometric samplesinthehigh-redshiftbins(Mylesetal.2021).

TheSOMPZmethodincorporatesmethodsforassessingthelike- lihoodL[PZ|n_u(z)]ofobtainingthevariousSOMPZdataelements (SOM cell counts, etc.) given a candidate set of nu(z) redshift distributionsforthetomographicbins,whichaccountforshotnoise and sample variance in the various catalogues usedby SOMPZ.

Theconstructionof thislikelihood andthe methodsforsampling candidaten(z)distributionsfromitaregivenbyS´anchez&Bern- stein(2019).Potentialselectionbiasesinthespectroscopicredshift

Figure4. RedshiftdistributionoftheBOSS/eBOSSsampleindata.

assignmentsareestimatedbycompilingn(z)realizationsobtained by calibratingwiththreedifferentsetsofspectroscopic/multiband photometricsamples.Redshiftuncertaintiesrelatedtothezero-point calibrationareaddedaftertheSOMPZrealizationsareinformedby the clusteringmeasurements(Mylesetal.2021).Thisisdonefor efficiencyreasonsanditdoesnotaffectthemainresultsofthiswork.

TheSOMPZ process is completelyreproduced in simulations, including the creation of spectroscopic catalogues from small- area surveys, but thesesimulations do not take into account the uncertainties related to unknown redshift selection biases in the spectroscopic/multibandsamples.Asaresultoftheslightdifferences ofthesimulatedY3sourcesampledataequivalent,thebinedgesin theequivalentBuzzardcatalogueare[0.0,0.346,0.628,0.832,2.0].

Estimatesofthen(z)obtainedinsimulationsareshowninFig.5.

4 R E S U LT S O N S I M U L AT I O N S A N D S Y S T E M AT I C E R R O R S

Inthissection,wepresenttheresultsofourtwocalibrationstrategies performedinsimulations.Inparticular,weaimtoevaluatethesys- tematicuncertaintiesofeachmethod,andverifythatthecalibration procedureinsimulationsworksasexpected.Notethatatnopoint arethesimulationsusedtomakecorrectionstothedata;ratherthe simulationsareusedto(1)estimatethelevelofuncertaintytoassign tovarioussystematicerrors,and(2)validatethatthemethodyields resultsforn(z)consistentwithtruth.

Beforefocusingonthedetailsofthetwocalibrationprocedures, we show in Fig. 6 the redshift distributions estimates obtained using the clustering-z nu(z) estimator (following equation 8) on simulations,comparedtothetruedistributions.Theangularscales consideredintheclusteringmeasurementshavebeenchosentospan thephysicalintervalbetween1.5and5.0Mpc.Thesebounds(which areappliedtothedataaswell)areselectedsothattheupperbound isbelowtherangeusedforthew(θ)statisticsusedincosmological analyses,thusallowingtheclustering-zlikelihoodstobeessentially statisticallyindependentofcosmology,andpermittingustoproduce n(z)samplesinanMCMCchainthatrunsbefore,andindependent of,thecosmology.Thevaluesofbrintheclustering-zanalysisare notrequiredtomatchthoseusedinthecosmologicalanalyses.The lowerboundischosentoproducehighsignal-to-noiseratioS/Nwhile mitigatingfailuresofthelinearbiasmodel.

Westartwithanidealizedcase:thedistributionsshowninFig.6are obtainedusingredMaGiCgalaxiesasareferencebinnedusingtrue redshift.Insimulationswealsohaveanaccurateestimateofb_u(z), obtainedfromtheautocorrelationsofeachofthetomographicbinsof

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Figure5. SOMPZredshiftdistributions,asestimatedinsimulations(upperpanels)andindata(lowerpanels),forthefourtomographicbinsconsideredinthis analysis.Thebandsrepresentthe68percentconfidenceintervalspannedbytheSOMPZn(z)realizations.

Figure6. Sourcesredshiftdistributionsestimatedusingclustering-zinsimulationsforanidealizedset-up(seethetextinSection4),comparedtothetruth (blacklines).Thetoppanelsshowtheredshiftdistributions;themiddlepanelsshowtheratiobetweenthetruen(z)andthen(z)estimatedusingclustering-z;

andthebottompanelsshowthemeanoftheredshiftdistributions.Theredlinesrepresenttheclustering-zestimatesobtainedusingtheestimatorintroduced byequation(8).Thebluelinesrepresenttheclustering-zestimatedobtainedfurthercorrectingforthetermbu,whichisonlypossibleinsimulations.The fourdifferenttomographicbinsusedintheDESY3cosmologicalanalysisareshown.WeusedredMaGiCgalaxiesasthereferencesample,binnedusingtrue redshifts.Forthisplot,wealsosubtractedfromtheclustering-zn(z)estimatestheexpectedmagnificationcontributioninsimulations(AppendixA);thishas onlyamildeffectathighredshift(z>0.6)forthefirsttwobins.Theredshiftdistributionsarenormalizedoverthesameinterval.Thegreyshadedregions indicatetheintervalconsideredforthemeanmatchingmethod.Themeanofthedistributionsshowninthebottompanelsiscomputedonlyconsideringthegrey intervals.Errorbarsonlyincludestatisticaluncertainties.

theunknownsample,dividedintothinbinsofwidthz=0.02.²This isnotpossibleindatasincetheprecisionofthephotometricredshift

2Inordertomeasuretheautocorrelations,wegeneratedrandomsproperly accountingfortheWLmask.WealsocreatedsystematicweightsfortheWL sampleusingthesameprocedureusedforredMaGiCgalaxies(althoughwe foundtheyhaveanegligibleimpact).

isnotsufficienttodividethesampleinbinsofadequatewidth.Fig.6 showstheimpactontheestimatedn(z)’sofassumingweknowb_u(z) withgoodaccuracy(incyan),dividingequation(8)bybu(z). We notethatcorrectingforb_udrivesboththeshapeofthedistributions andthemeanvalueclosertothetruth,whichareotherwisebiased.

Aswecannotestimateb_uindata,thishighlightsthatvariationinb_u introducesasystematicuncertaintythathastobequantified.Note thattheerrorsbarsinFig.6onlyincludestatisticaluncertainties.