Towards an understanding of third-order galaxy-galaxy lensing

0.23 MB·English

by  Patrick Simon

Astronomy&Astrophysicsmanuscriptno.understandingG3L c ESO2012
(cid:13)
February10,2012
Towards an understanding of third-order galaxy-galaxy lensing
PatrickSimon,PeterSchneider&DanielaKu¨bler
Argelander-Institutfu¨rAstronomie,Universita¨tBonn,AufdemHu¨gel71,53121Bonn,Germany
2
e-mail:[email protected]
1
0
ReceivedFebruary10,2012
2

b ABSTRACT
e
F
Context.Third-order galaxy-galaxy lensing (G3L) is a next generation galaxy-galaxy lensing technique that either measures the
9 excessshearaboutlenspairsortheexcessshear-shearcorrelationsaboutlenses.Fromtheirdefinitionitisclearthatthesestatistics
assessthethree-pointcorrelationsbetweengalaxypositionsandprojectedmatterdensity.
] Aims.Forfutureapplicationsofthesenovelstatistics,weaimatamoreintuitiveunderstandingofG3Ltoisolatethemainfeatures
O thatpossiblycanbemeasured.
Methods.Weconstructatoymodel(“isolatedlensmodel”;ILM)forthedistributionofgalaxiesandassociatedmattertodetermine
C
themeasured quantitiesofthetwoG3L correlationfunctionsandtraditionalgalaxy-galaxy lensing(GGL)inasimplifiedcontext.
h. The ILM presumes single lens galaxies to be embedded inside arbitrary matter haloes that, however, are statisticallyindependent
p (“isolated”)fromanyotherhaloorlensposition.Clustersofgalaxiesandtheircommonclustermatterhaloesareaconsequenceof
- clusteringsmallerhaloes.Inparticular,theaveragemass-to-galaxynumberratioofclustersofanysizecannotchangeintheILM.
o Results.GGLandgalaxyclusteringalonecannotdistinguishanILMfromanymorecomplexscenario.Thelens-lens-shearcorrelator
r incombinationwithsecond-orderstatisticsenablesustodetectdeviationsfromaILM,though.Thiscanbequantifiedbyadifference
t
s signaldefinedinthepaper.WedemonstratewiththeILMthatthiscorrelatorpicksuptheexcessmatterdistributionaboutgalaxypairs
a insideclusters,whereaspairswithlenseswellseparatedinredshiftonlysuppresstheoverallamplitudeofthecorrelator.Theamplitude
[ suppressioncanbecalibrated.Theshear-shear-lenscorrelatorissensitivetovariationsamongmatterhaloes.Inprinciple,itcouldbe

devisedtoconstraintheellipticitiesofhaloes,withouttheneedforluminoustracers,ormaybeevenrandomhalosubstructure.
1
v Keywords.Gravitationallensing:weak–Galaxies:halos–(Cosmology:)large-scalestructureofUniverse
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2
9
1. Introduction
1
2. Gravitationallensing(Schneider2006,forarecentreview)hasestablisheditselfasvaluabletoolforcosmologytoinvestigatethe
0 large-scale distribution of matter and its relation to visible tracers such as galaxies. In the currently favoured standard model of
2 cosmology(e.g.,Peebles1993;Dodelson2003),the majorfractionofmatteris a non-baryoniccolddarkmattercomponent,i.e.,
1 matterwithnon-relativisticvelocitiesduringtheepochofcosmicstructureformation.Inthissituation,gravitationallensingisan
: excellentprobeasitissensitivetoallmatteraslongasitinteractsgravitationally.
v
i Themainobservableinlensingisthedistortionofshapesofgalaxyimages,intheweaklensingregimemainly“shear”,bythe
X interveninginhomogeneousgravitationalpotentialthat is traversedby light bundlesfromthe galaxy.Over the course of the past
r decade,applicationsofthegravitationallensingeffecthavecomeofage.Tonameafewresults(Bartelmann2010,andreferences
a
therein),itwasusedtomapthedarkmatterdistribution,tostudythematterdensityprofilesingalaxyclustersandtodeterminetheir
masses, to measure the relationbetweenthe galaxyand darkmatterdistribution(the so-calledgalaxybias), to constrainthe total
matterdensityoftheUniverseanditsfluctuationpowerspectrum,andveryrecentlytogatherindependentevidencefortheoverall
acceleratedexpansionofthecosmos(Schrabbacketal.2010).
Of particular interest for this paper is the so-called galaxy-galaxy lensing technique where positions of foreground galaxies
(“lenses”)arecorrelatedwiththe(weaklensing)shearonbackgroundgalaxyimages(“sources”).Thereby,statisticalinformation
on the projected matter distribution around lens galaxies can be extracted. Since the first attempt by Tyson et al. (1984) and the
firstdetection(Brainerdetal.1996;Griffithsetal.1996)ofthiseffect,galaxy-galaxylensingnowadaysisawidelyappliedrobust
methodtostudythegalaxy-matterconnection(Fischeretal.2000;McKayetal.2001;Guzik&Seljak2002;Hoekstraetal.2002;
Pen et al. 2003;Hoekstra et al. 2004;Seljak & Warren 2004;Sheldon et al. 2004;Mandelbaumet al. 2005;Kleinheinrich et al.
2005;Mandelbaumetal.2006b,a;Simonetal.2007;Parkeretal.2007;vanUitertetal.2011).Thetraditionalandhithertomainly
employedapproachistocorrelatethepositionofonelenswiththeshearofonesourcegalaxy(GGLhereafter).
Schneider & Watts (2005) advanced the traditional GGL by considering a new set of three-point correlation functions that
eitherinvolvestwolensesandonesource(“correlator ”)ortwosourcesandonelens(“correlatorG ”).Bothcorrelatorsaretools
to directly study higher-order correlations between gaGlaxies and the surrounding matter field. In th±e literature, this technique is
termed3rd-ordergalaxy-galaxylensingorgalaxy-galaxy-galaxylensing.Therearealternativebutmathematicallyequivalentways
toexpressthesestatistics,e.g.,theaperturestatistics 2M and M2 insteadofthethree-pointcorrelationfunctionsG and
(Schneider & Watts 2005; Simon et al. 2008). InhpNracticapail meahsNuremapeints usually the aperture statistics are preferred, as±they
aGutomaticallyremoveunconnected2nd-ordercontributionsin estimatorsof the statistics and allow oneto separateE-modesfrom
1
Simonetal.:TowardsanunderstandingofG3L
B-modesorparitymodes,ofwhichthelattertwocannotbegeneratedbygravitationallensingasleadingordereffect.Inthispaper,
wefocusontheE-modesinthecorrelators andG .
G3LhasalreadybeenmeasuredinconteGmporary±lensingsurveys,suchastheRed-SequenceClusterSurvey(Simonetal.2008),
andwillthereforepresumablyberoutinelymeasuredwithongoingsurveyssuchasKiDS1,Pan-STARRS2,DES3,orinthefuture
surveysEuclid4andLSST5.Theprospectsoflearningmoreonthegalaxy-matterrelationornewobservationaltestsfortheoretical
galaxymodels(e.g.Weinbergetal.2004;Boweretal.2006;DeLuciaetal.2007)withG3Larethusquitepromising.
The main obstacle for exploiting the new G3L statistics is their physical interpretation. It is clear from the definition that
G
quantifiestheshearsignal(orprojectedmatterdensity)inexcessofpurelyrandomlydistributedlenses,pickinguponlysignalfrom
clusteredlenspairs(e.g.,Johnston2006),andthatG isatwo-pointcorrelationfunctionofshearassociatedwithmatterphysically
closetolenses.Itisunclear,however,whatphysical±informationthistranslatestoandwhatnewfeaturemaybecontainedinG3L
thatmaybemissingorisdegenerateintraditionalGGL.Toelucidatethesenewstatisticsandtopavethewayfornewapplicationsof
G3L,weconceivehereasimplisticmodelforthedistributionoflensesandmatter:theisolatedlensmodel.ThenG3Lisflashedout
inthelightofthismodel.Forthedefinitionofquantitiesrelevantforweakgravitationallensing,wereferthereadertoBartelmann
&Schneider(2001).ThemathematicalmachineryofahalomodelexpansiondevisedinthecalculationsisverysimilartoScherrer
&Bertschinger(1991),althoughusedinadifferentphysicalcontext.
Thestructureofthepaperlaysoutasfollows.Sect.2introducesourmodelandderivesthetangentialshearaboutalens,theGGL
signal,expectedfromthis description.Sect. 3 movesonto calculate thelens-lens-shearor correlatorfor thisspecific scenario.
G
Sect. 4 does the same for the shear-shear-lensorG correlator. The final Sect. 5 summarises the main conclusionsdrawn in the
precedingsections. ±
2. Theisolatedlensmodelandgalaxy-galaxylensing
Herewelayoutasimplemodelforthedistributionofmatterandgalaxiesinsideit.Thismodelisfoundedontheassumptionthat
lensesareembeddedinsideamatterhalothatgeneratestheshearprofileγ (θ;α)actinguponabackgroundsource,whereθisthe
h
separationvectorfromthe centroidof the galaxy,and α denotesa set of intrinsic haloparametersthat controlthe matter density
profileof thehalo.Importantly,the intrinsic parametersare statistically independentofthe intrinsichalo parametersof anyother
haloortheseparationsofotherlenses.Wehencecoinlensesandtheirhosthaloesinthisscenario“isolated”.Inthissense,thisisa
verycrudehalomodelrepresentation(Cooray&Sheth2002)ofthelensandmatterdistribution,assumingforsimplicitythatevery
haloisoccupiedbyexactlyone(lens)galaxy.Noticethatmatterwhichisstatisticallyindependentofthelensesdoesnotneedtobe
accountedfor,asthiswouldnotcontributetoagalaxy-mattercross-correlationfunction,althoughitcertainlywouldaffectthenoise
inameasurement.
Inthefollowing,2Dpositionsontheflatskyare,forconvenience,denotedbycomplexnumbersθ = θ +iθ whereθ (θ )is
1 2 1 2
the position in directionof the x(y)-axis.By θ = θ = √θθ we denotethe modulusof θ. Likewise the Cartesian shear 2-spinor
γ =γ +iγ isdenotedascomplexnumber.Wede|fi|netheta∗ngential,γ,andcross,γ ,shearofγ (θ)relativetotheoriginby
c 1 2 t c
×
γ(θ;ϕ):=γ(θ)+iγ (θ)= e 2iϕγ (θ), (1)
t − c
× −
whereϕisthepolarangleofθ.
2.1.Isolatedlensmodel
Withintheisolatedlensmodel(ILMhereafter),thenumberdensitydistributionoflensesonthe(flat)skywithareaAis
Nd
n (θ)= δ(2)(θ θh), (2)
g D − i
Xi=1
whereδ(2)(θ)istheDiracdeltafunction,andtheresultingshearfieldis
D
Nd
γ (θ)= γ (θ θh;α), (3)
c h − i i
Xi=1
stickingashearprofiletoeveryoftheN lenspositionθh.Theshearprofileisdirectlyrelatedtotheprojectedmatterdensityabout
d i
thelens.Inthefollowingwewillusethelensnumberdensitycontrast
n (θ)
κ (θ):= g 1, (4)
g
n −
g
1 http://www.astro-wise.org/projects/KIDS/
2 http://www.cfht.hawaii.edu/Science/CFHLS/
3 http://www.darkenergysurvey.org
4 http://sci.esa.int/euclid, seealsoLaureijsetal.(2011)
5 http://www.lsst.org/
2
Simonetal.:TowardsanunderstandingofG3L
unclustered clustered
Fig.1. Illustrationof theisolatedlensmodel. Galaxiesaredepictedby blackpixels,their matterhaloesasreddisks. Forsimplicityallhaloes
areidenticalinthisvisualisation.Leftpanel: Lensesaredistributedrandomlyonthesky.Rightpanel:Lensesclustertoproduceclumpswitha
commonmatterenvelope, yetstilldescribedassumsofindividual haloes.Onthestatisticallevel,thisclumpingisquantifiedbynon-vanishing
2nd-orderand33d-ordercorrelationfunctionsωandΩ,respectively.
wheren := N /AisthemeannumberdensityoflenseswithintheareaA.Forthe2nd-orderangularclusteringcorrelationfunction
g d
oflensesonthesky(e.g.Peebles1980),weemploythefunction
ω(θ )= κ (θ )κ (θ ) (5)
12 g 1 g 2
| | (cid:28) (cid:29)
withθ :=θ θ beingtheseparationvectoroftwopositions.Avalueω(θ )>0expressesanexcessofgalaxypairsatseparation
ij i j 12
−
θ comparedtoapurelyrandomdistribution.
12
The ILM is a more general description than may appear at first sight: Galaxy clumps with joint matter envelopes (“galaxy
clusters”) are not explicitly excluded, although every individual halo does host only one lens. To form matter haloes of whole
clusters,wecanalwayssticktogetherandoverlapmatterhaloes,changingtheclusteringcorrelationfunctionsinconsequence.The
differencebetweentheleftandrightpanelofFig.1liesthereforeinthechoiceoftheclusteringcorrelationfunctions,whichwill
enter the following calculations. Hence, the ILM expands larger matter haloes as sums of individual matter haloes of clustered
galaxies.Crucially,however,the ILMisincapabletoimplementclumpsof N galaxiesthatcontainonaveragemoremassthan N
isolatedgalaxies;themeanmatter-to-lens-numberratiohastobeconstantthroughout.Toformaclumpwithahighermass-to-lens
numberratiowouldrequiretoincreasethemassofallindividualhaloessimultaneously,i.e.,tochangetheinternalhaloparameters
of all lenses inside the clump in a similar fashion. This is notallowed, except by chance, however,since internalparametersare
statistically independent.A generalisation of the ILM in this direction could be achieved in a full-scale halo model with matter
haloes hosting more than one galaxy. Note that the luminosity of a lens can also be seen as an internal parameter. Therefore, a
constant mass-to-lens number ratio plus statistical independenceof internal parameters amounts also to a constant mass-to-light
ratioofallclumpsinthemodel.
2.2.Galaxy-galaxylensing
Before we embarkon 3rd-orderstatistics, we start with the morefamiliar 2nd-orderGGL (Bartelmann& Schneider2001).These
statisticsarethemeantangentialshearγ aboutalensatseparationϑ= θ θ (Fig.2),definedbythecorrelator
t | 2− 1|
n (θ )γ (θ ) = e+2iϕn¯  κ (θ )γ(θ ;ϕ) + γ(θ ;ϕ) = e+2iϕn¯ γ(ϑ). (6)
(cid:28)Owgin1gtocis2ot(cid:29)ropy−,themegan(cid:28)tgang1entia2lshe(cid:29)arγ|(cid:28) i s {=o2z0n l y }(cid:29)afun−ctionofgsetparationϑandindependentofthepolarangleϕ.Thebracket
t
... denotestheensembleaverageoverlensnumberdensitiesandshearconfigurations.Theunderbracedtermhastovanishdueto
h i
thestatisticalisotropyandhomogeneity.
As lens numberdensities and shear configurationsare expandedin terms of haloesin the ILM, we consider the correlatoras
ensembleaverageoverallpossiblelenspositionsandinternalhaloparameters:
Nd Nd Nd
n (θ )γ (θ ) = δ(2)(θ θh)γ (θ θh;α ) = δ(2)(θ θh)γ (θ θh;α) + δ(2)(θ θh)γ (θ θh;α ) , (7)
(cid:28) g 1 c 2 (cid:29) (cid:28)iX,j=1 D 1− i h 2− j j (cid:29) Xi= 1 (cid:28) D 1 − i h 2 − i i (cid:29)i i,X j = 1 (cid:28) D 1 − i h 2 − j j (cid:29) i,j
one halo two halo
− −
| {z } | {z }
3
Simonetal.:TowardsanunderstandingofG3L
j j 1(j +j )
θ2 2 j 2 2 1 2
1
θ θ
θ 2 j
2
1 1
ϑ
θ
J f J J 1
ϕ 2 3 1 2 f J
3
1
θ
θ1 3 θ
3
Fig.2. Left panel: Illustration of the parametrisation of the GGL correlator γ(ϑ). The lens is located at θ , the source is at θ . Middle and
t 1 2
rightpanel:IllustrationoftheparametrisationoftheG3Lthree-pointcorrelatorsG (ϑ ,ϑ ,φ )(middle),andthegalaxy-galaxy-shearcorrelation,
1 2 3
±
(ϑ ,ϑ ,φ )(right).ThefiguresarecopiedfromSchneider&Watts(2005).
G 1 2 3 e
e
whichsplitsintotwoseparatesumswithensembleaveragesoverallhaloparametersofonehalo(one-haloterm)
1
... := d2θh ... (8)
(cid:28) (cid:29)i AZ i (cid:28) (cid:29)αi
ortwohaloes(two-haloterm)
1
... := d2θhd2θh 1+ω(θh θh) ... . (9)
(cid:28) (cid:29)i,j A2 Z i j (cid:16) | i − j| (cid:17)(cid:28) (cid:29)αi,αj
Thestatisticalindependenceofhalopositionsθhi andinternalhaloparametersαiisexplicitlyusedhere;h...iαi andh...iαi,αj arethe
averagesoverinternalhaloparametersofasinglehaloorjointlyfortwohaloes,respectively.Forthelatter,westressagainthatwe
willassumestatisticalindependenceofα andα .
i j
Inthefollowing,wewillneedtheaveragehaloshearprofile
γ (θ):= γ (θ;α) = dαP (α)γ (θ;α)= e+2iϕγ (θ), (10)
h (cid:28) h (cid:29)α Z α h − t,h
P (α)istheprobabilitydensitydistributionfunction(p.d.f.)oftheinternalhaloparametersα.Thetangentialhaloshearprofileγ
α t,h
isnottobeconfusedwithγ inEq.(6)thatdescribesthetotalmeantangentialshearaboutalensincludingcontributionsfromthe
t
lens halo and haloes of clustering neighbouringlenses. Due to rotational symmetry,the average profile γ has a vanishing cross
h
shearcomponent,forwhichreasonwecanexpressitintermsofthetangentialhaloshearfunctionγ (θ),whichisonlyafunction
t,h
oftheseparationθ.
Utilisingthisdefinition,wearrivefortheGGLcorrelatorat:
Nd 1 Nd 1
n (θ )γ (θ ) = γ (θ ;α) + d2θ 1+ω(θ θ ) γ (θ;α) (11)
(cid:28) g 1 c 2 (cid:29) Xi=1 A(cid:28) h 21 (cid:29)α i,Xj=1 A2 Z ′(cid:2) | ′− 21| (cid:3)(cid:28) h ′ (cid:29)α
n¯ γ (θ )+n¯2 d2θ ω(θ θ )γ (θ )+n¯2 d2θ γ (θ ) .
≈ g h 21 gZ ′ | ′− 21| h ′ g Z ′ h ′
=0
| {z }
Theunderbracedtermmustvanishduetoradialsymmetry.Insidethesumsalltermsbecomeindependentofindividuallenspositions
θh andhaloparametersα duetotheaveraging.Thelaststepassumesthatthenumberofhaloesislarge,i.e.,N 1,inparticular
i i d ≫
N (N 1) N2.Thiswillalsobeassumedforallfollowingcalculations.
d d− ≈ d
Employing(6)and(10),wefinallyfind:
2π
γt(ϑ)=γt,h(ϑ)+n¯g dθ′θ′dϕ′e2i(ϕ′−ϕ)ω(Ψ)γt,h(θ′)=γt,h(ϑ)+n¯g ∞dθθγt,h(θ) dϕcos(2ϕ)ω(Ψ) (12)
Z Z Z
0 0
withtheexpression
Ψ:= θ2+ϑ2 2θϑcosϕ. (13)
q −
Thelaststepin(12)exploitsthatω(θ)hasvanishingimaginarypart.InthespecificILMdescription,theaverageshearisexpanded
intermsoftwocomponents:ThefirstterminEq.(12)isthe one-haloterm,dominatingatsmallseparations,whereasthesecond
term is the two-halo term due to the clustering of haloes. Importantly, GGL is only sensitive to the average lens halo γ (θ) but
h
insensitivetodeviationsofγ (θ;α)fromγ (θ)intheactualhalopopulation,whichareexplicitlyallowedwithintheILM.
h h
4
Simonetal.:TowardsanunderstandingofG3L
3. Lens-lens-shearcorrelator
Wenowturntothe3rd-ordergalaxy-galaxylensingstatistics,startingwiththeconstellationoftwolensesandonesourceasdepicted
intherightpanelofFig.2.Thecorrelatorconsidersacross-correlationbetweenlensnumberdensitiesattwopositionsθ andθ
1 2
andtheshearatθ :
3
(cid:28)ng(θ1)ng(θ2)γc(θ3)(cid:29)=−n¯2ge+i(ϕ1+ϕ2)G(ϑ1,ϑ2,φ3). (14)
e
Statisticalhomogeneityandisotropyimpliesthatwecanextractacorrelationfunction ˜ fromthecorrelatorthatissolelyafunction
G
oflensandsourceseparations,ϑ andϑ ,andtheopeningangleφ .Thisfunctionissplitintoanunconnected, ,andaconnected
1 2 3 nc
G
part
e
ϕ +ϕ
(ϑ ,ϑ ,φ )= κ (θ )κ (θ )γ θ ; 1 2 = (ϑ ,ϑ ,φ ) (ϑ ,ϑ ,φ ). (15)
1 2 3 g 1 g 2 3 1 2 3 nc 1 2 3
G (cid:28) (cid:18) 2 (cid:19)(cid:29) G −G
e e
Theconnectedpartvanishesforunclusteredlenses,whiletheunconnectedpartcanbeshowntobegenerally(Schneider&Watts
2005):
Gnc(ϑ1,ϑ2,φ3):=e−iφ3γt(ϑ1)+e+iφ3γt(ϑ2), (16)
wehichisjustthesumoftheGGLshearprofilearoundeachlens.Therefore, encodestheshearinexcessofwhatisexpectedfrom
G
unclusteredlenseswiththeaverageshearprofileγ aroundthem,or:Itquantifiestheshearsignalaboutclustered lenspairs.Note
t
thattheunconnectedtermsdonotcontributetotheaperturestatistics 2M andarethusnottheprimaryquantitymeasuredwith
ap
hN i
G3L.
3.1.Derivation
For withintheILM,weneedtoevaluatetheconnectedtermsofthecorrelator n (θ )n (θ )γ (θ ) withthemodelspecificsEqs.
g 1 g 2 c 3
G h i
(2)and(3):
Nd
n (θ )n (θ )γ (θ ) = δ(2)(θ θh)δ(2)(θ θh)γ (θ θh;α ) (17)
(cid:28) g 1 g 2 c 3 (cid:29) (cid:28)i,Xj,k=1 D 1− i D 2− j h 3− k k (cid:29)
Nd Nd
= δ(2)(θ θh)δ(2)(θ θh)γ (θ θh;α) + δ(2)(θ θh)δ(2)(θ θh)γ (θ θh;α) +(2 perm.)
Xi= 1 (cid:28) D 1 − i D 2− i h 3 − i i (cid:29)i i,X j = 1 (cid:28) D 1 − i D 2 − j h 3 − i i (cid:29) i , j
one halo two halo
− −
|Nd {z } | {z }
+ δ(2)(θ θh)δ(2)(θ θh)γ (θ θh;α )
i,X j , k = 1 (cid:28) D 1 − i D 2− j h 3 − k k (cid:29) i , j,k
three halo
−
| {z }
where
1
... := d2θhd2θhd2θh 1+ω(θh)+ω(θh)+ω(θh )+Ω(θh,θh ,θh) ... (18)
(cid:28) (cid:29)i,j,k A3 Z i j kh i j i k j k ik jk ij i(cid:28) (cid:29)αi,αj,αk
unconnected
| {z }
istheensembleaverageoverthreehaloes(three-haloterm).By
Ω(θ ,θ ,θ )= κ (θ )κ (θ )κ (θ ) (19)
13 23 12 g 1 g 2 g 3
(cid:28) (cid:29)
we denote the (connected) 3rd-order angular clustering correlation function of the lenses that only depends on relative galaxy
separations. The sum (17)hence decaysinto a one-haloterm (first sum), two-halo(nextsum plustwo identical sums apartfrom
permutationsof the indicesi and j) andthe three-haloterm (lastsum). The one-haloterm vanishesfor θ ,θ , i.e., distinct lens
1 2
positions,owingtotheDeltafunctions.Fortheconnectedterms,inthehalocorrelators ... or ... onlythesummandswith
i,j i,j,k
the leading order clustering correlation functions are relevant, i.e., only the terms withhΩiin thehcoirrelator ... , while terms
i,j,k
h i
withωor1arepartoftheunconnectedterms(underbraced).Theybelongtotheunconnectedpartofthe3rd-orderlensclustering.
Similarly,in ... onlytermsassociatedwithωareofrelevance,thetermsgeneratedby1,theunconnectedpartofthe2nd-order
i,j
h i
lensclustering,willgointo .
nc
ByevaluationofallthesGeensembleaveragesonetherebyobtainsforθ ,θ ,θ :
1 2 3
e
(ϑ ,ϑ ,φ )= (ϑ ,ϑ ,φ )+ (ϑ ,ϑ ,φ ) (20)
1 2 3 2h 1 2 3 3h 1 2 3
G G G
5
Simonetal.:TowardsanunderstandingofG3L
withthetwo-haloterms(weutilisetherelationγ ( θ)=γ (θ)followingfromEq.10)
h − h
G2h(ϑ1,ϑ2,φ3) := −e−i(ϕ1+ϕ2)ω(|θ1−θ2|) γh(θ3−θ1)+γh(θ3−θ2)
(cid:16) (cid:17)
= −e+i(ϕ1+ϕ2)ω(ϑ2) −e+2iϕ1γt,h(ϑ1)−e+2iϕ2γt,h(ϑ2)
(cid:16) (cid:17)
= ω(ϑ3) e−iφ3γt,h(ϑ1)+e+iφ3γt,h(ϑ2) , (21)
(cid:16) (cid:17)
thelens-lensseparation
ϑ = ϑ2+ϑ2 2ϑ ϑ cosφ (22)
3 q 1 2− 1 2 3
andthethree-haloterm
G3h(ϑ1,ϑ2,φ3) := −nge−i(ϕ1+ϕ2)Z d2θΩ |θ1−θ|,|θ2−θ|,|θ1−θ2| γh(θ3−θ)
(cid:16) (cid:17)
= −nge−i(ϕ1+ϕ2)Z d2θΩ |θ13+θ|,|θ23+θ|,ϑ3 γh(θ)
(cid:16) (cid:17)
= ngZ dθθdϕΩ Υ(ϑ1,θ,ϕ−ϕ1),Υ(ϑ2,θ,ϕ−ϕ2),ϑ3 e−i(ϕ1+ϕ2−2ϕ)γt,h(θ)
(cid:16) (cid:17)
= ng dθθdϕΩ Υ(ϑ1,θ,ϕ+φ3),Υ(ϑ2,θ,ϕ),ϑ3 e+2i(φ3+ϕ)γt,h(θ)
Z
(cid:16) (cid:17)
2π
= nge+2iφ3 d∞θθγt,h(θ) dϕcos(2ϕ)Ω Υ(ϑ1,θ,ϕ+φ3),Υ(ϑ2,θ,ϕ),ϑ3 , (23)
Z Z
0 0 (cid:16) (cid:17)
forwhichwehaveintroducedtheauxiliaryfunction
Υ(θ ,θ ,φ):= θ2+θ2+2θ θ cosφ. (24)
1 2 1 2 1 2
q
Thetransformationsin(23)useφ =ϕ ϕ andachangeoftheintegralvariablesϕ ϕ+ϕ andθ θ+θ .Thelaststeputilises
3 2 1 2 3
thatΩ(...)hasavanishingimaginarypa−rt. 7→ 7→
3.2.Interpretation
Inthecontextof ,wecandefineaexcessmassmapinthefollowingway.Thefunction can,forlenspairsoffixedseparationϑ ,
3
bemappedasexcGessshearfieldatpositionθ withCartesianshearvalue G
3
θ θ
γ (θ θ ,θ )= 13 23 (θ ,θ ,φ ) , (25)
c 3| 1 2 − θ θ G 13 23 3
13 23
| || |
whereφ isdefinedasanglespannedbyθ andθ .Inthismap,wefixthelenspositionsatθ = +ϑ /2andθ = ϑ /2onthe
3 23 13 1 3 2 3
−
x-axis.Thisshearmapcanbeconvertedintoaconvergencemap(e.g.Kaiser&Squires1993),asforexampledoneinSimonetal.
(2008).Theexcessmassfromthe two-halotermof theILMis justthehalomassabouteachlensatθ andθ , weighedwith the
1 2
clusteringstrengthω(ϑ )ofthepair.Thisisexactlythemasswewouldanticipatearoundapairoflenses,afteronehassubtracted
3
themassaroundunclusteredpairs(Eq.21withω(ϑ ) 1)andifoneignoredtheeffectofthirdhaloes.Theexcessmassstemming
3
≡
fromathirdhalo,clusteringaroundthelenspair,isdescribedbythethree-haloterm.
That the excess shear or mass originates from galaxy clusters can be argued from the ILM (Fig. 3). In the ILM, the excess
shearisexpressedintermsofthe2nd-and3rd-orderlensclusteringcorrelationfunctionsonthesky.Wehavenosignal,iflensesare
unclustered,i.e.,ω=Ω=0,orsimplyifwehavenogalaxyclusters.Unclusteredlenseswithstatisticallyindependentmatterhaloes
cannotproduceanyexcessmass.Ontheotherhand,theystillmaygenerateaGGLsignal(12)ifγ ,0.Iflensescluster,i.e.,ω,0
h
or Ω,0, we will get automatically non-vanishing contributions to . For lens-lens separations comparable or smaller than the
G
typicalangularsizeofacluster,mostlenspairswillinhabitclustersandcontributemostlytotheexcessmass.Therefore,thosepairs
probeessentiallythematterenvironmentofclusters,providedtheyareatsimilarredshift.Wedonotexpectrelevantcontributionto
frompairsoflenseswithdistinctredshifts(apparentpairs),though.Imagineacatalogueoflensesinwhichallgalaxiesareclearly
G
separatedinradialdistance.Onthesky,theselensesare(a)unclusteredand(b)theirmatterenvironmentsaremutuallystatistically
independentowingtothelargephysicaldistancesbetweenlenses.Thisexactlycoverstheaforementionedsituationasreflectedin
aILMwithvanishingωandΩ: fromthislenscataloguevanishes.Incomparisonwithasampleofgalaxiesallatsimilarradial
G
distance,asurveywithradialspreadinthelensdistributionwillhavealargerfractionofapparentpairs,areducedangularclustering
oflensesandhenceaoverallsuppressedamplitudeof .Thissuppressioncanbecorrectedfor,iftheradialdistributionoflensesis
G
specified(AppendixA).
TheILMisonlyanapproximationfortheclusteringofgalaxiesandmattersinceachangeofthematter-to-lightratiowithsize
of structures is not possible. Every structure can only be a sum of individual haloes with no correlation to each other. Contrary
6
Simonetal.:TowardsanunderstandingofG3L
Fig.3. Rightpanels:Excessmassaroundlenspairswithfixedseparation;squaresindicatethelenspositionsinsidethemaps.Leftpanels:Excerpts
oftheunderlyingILMmockdata:Lenses,shownaslittledots,areeitherclustered(topleft)orrandomlydistributedonthesky(bottomleft).For
simplicity,everylenshasthesameindividualmatterhalowithaGaussianlensingconvergenceprofile(r.m.s.sizeis20pixel)stickedtoit.The
clusteredlenshaloesproducesthejointmatterhaloofgalaxyclustersinthismodel.Theintensityscaleintheleftpanelsdepictsthecombined
lensingconvergenceofalllenses;thisisprobedasshearbyasampleofsourcegalaxies.Notethattheangularscaleortheshearamplitudeareof
noparticularinteresthere.Thebottomrightpanelistheactualmeasurementofthebottomleftscenariowiththecolourscaleofthetoprightpanel.
to GGL thislimitation is relevantfor G3L,as discussed in the following.We first notice thatin comparisonwith GGL, the lens-
lens-shearcorrelatorunderILMassumptionsseeminglydoesnotprovideanyfundamentallynewinformationaboutthelens-matter
connection.Toclarifythis,theclusteringcorrelationfunctions(ω,Ω)canbedeterminedbytheobservablelensangulardistribution
withoutexploitingthegravitationallensingeffect.Then,byutilisingωandtheobservedmeantangentialshearγ,theaveragehalo
t
shear profile γ can be constrainedfrom Eq.(12)so that all essential ingredientsfor predicting , Eq. (20), are already fixed. In
h G
particular, appearstobeonlysensitivetotheaveragehaloshearprofileasGGLis.Seenthisway,G3Lcanatmostcomplement
constraintsGon the mean shear profile γ . In a scenario more complexthan the ILM, however,this differs. Imaginethrowingin a
h
few completely different matter haloes into a ILM, hosting several galaxies simultaneously,that cannot be described as sums of
individualhaloes.ConventionalGGL wouldbe unableto detecta differenceto a ILMscenario,as we canstill use the clustering
correlation function ω and the GGL signal γ¯ with (12) to define an average lens shear profile γ¯ . Thereforethe ILM is always,
t h
even when falsely presumed,sufficientto consistently describe the GGL signal and the lens clustering.On the other hand, when
thencombinedwith , we wouldobserveinconsistencies,aswe failto correctlyexplain withEq.(20).Infact,thenewhaloes
G G
withmorethanonegalaxywouldproduceaone-haloterm(AppendixB),whichismissingintheILMdescription.Fromthiswe
concludethat , in combinationwith GGL, enablesus to detect whether a ILM sufficiently explainsthe data or whether a more
G
advanceddescriptionisrequired.
With the ILM as referencescenario we suggestto constructa test for the applicability of the ILM with the excesssignal ∆
G
constructedasfollows:
1. UseGGL,lensclusteringstatisticsandnumberstoobtaintheILMparameterset(n ,ω,Ω,γ¯ )viaEq.(12);
g h
2. DefinetheILMexcesssignalby
∆ (ϑ ,ϑ ,φ ):= (ϑ ,ϑ ,φ ) (ϑ ,ϑ ,φ ) (ϑ ,ϑ ,φ ), (26)
1 2 3 1 2 3 2h 1 2 3 3h 1 2 3
G G −G −G
wherethelasttwotermsonther.h.s.arethetwo-andthree-haloterm,Eqs.(21)and(23),fromtheILMdescription.
Avanishing∆ teststhevalidityofaILMdescriptionforthedataorexpressesthedeviationfromit.
G
7
Simonetal.:TowardsanunderstandingofG3L
4. Shear-shear-lenscorrelator
HerewepredictameasurementforthesecondG3LcorrelatorG ,giventhroughthecorrelationoftwoshearsandonelensnumber
density ±
e
γc(θ1)γc±(θ2)ng(θ3) =n¯−g1e+2i(ϕ1∓ϕ2)G (ϑ1,ϑ2,φ3). (27)
(cid:28) (cid:29) ±
e
The geometry of the correlator is depicted in the middle panel of Fig. 2. Here and in the following equations, a superscript “ ”
as in γ meansγ for γ and γ+ for the complexconjugateγ . This correlator measuresthe shear-shear correlationsas functi±on
c± c c− c c∗
oflensseparation.Asbeforewith ,symmetriesdemandthatthecorrelatordependsonlyonrelativeseparationsandanglesgiven
G
bythetriangledefinedbylensandsourcepositions.Itcontainsanunconnectedpartthatdescribestheshear-shearcorrelationsfor
randomlydistributedlenseswithnocorrelationtotheshearfield,namely(Schneider&Watts2005):
ϑ ϑ 4
Gn+c(ϑ1,ϑ2,φ3):=ξ+(ϑ3)e+2iφ3 ; Gn−c(ϑ1,ϑ2,φ3):=ξ−(ϑ3) ϑ32e+iφ3/2− ϑ13e−iφ3/2! ; (28)
e e
ϑ denotesthesource-sourceseparation.Asbefore,theunconnectedtermsdonotcontributetotheaperturestatistics,here M2 ,
3 hN api
and are thus of no particular interest for G3L. Subtracting the unconnected terms leaves us with the relevant excess shear-shear
correlationsaboutlenses,formally(cf.Eq.1)
G (ϑ ,ϑ ,φ )= γ(θ ;ϕ )γ (θ ;ϕ )κ (θ ) =G (ϑ ,ϑ ,φ ) Gnc(ϑ ,ϑ ,φ ). (29)
1 2 3 1 1 ± 2 2 g 3 1 2 3 1 2 3
± (cid:28) (cid:29) ± − ±
e e
4.1.Derivation
TheevaluationofG fortheILMboilsdowntoevaluatingtheconnectedtermsofthetriplecorrelator
±
γ (θ )γ (θ )n (θ ) (30)
c 1 c± 2 g 3
(cid:28) (cid:29)
Nd Nd
= δ(2)(θ θh)γ (θ θh;α)γ (θ θh;α) + δ(2)(θ θh)γ (θ θh;α )γ (θ θh;α )
X i (cid:28) D 3 − i h 1 − i i h± 2 − i i (cid:29)i iX, j , k (cid:28) D 3 − i h 1 − j j h± 2 − k k (cid:29) i , j,k
one halo three halo
− −
|Nd {z } | Nd {z }
+ δ(2)(θ θh)γ (θ θh;α )γ (θ θh;α) + δ(2)(θ θh)γ (θ θh;α)γ (θ θh;α )
Xi, j (cid:28) D 3 − i h 1 − j j h± 2 − i i (cid:29) i , j X i , j (cid:28) D 3 − i h 1 − i i h± 2 − j j (cid:29) i ,j
two halo 2
− −
|Nd {z }
+ δ(2)(θ θh)γ (θ θh;α)γ (θ θh;α) ,
Xi, j (cid:28) D 3 − j h 1 − i i h± 2 − i i (cid:29) i,j
two halo 1
− −
| {z }
which now contains a one-halo term, two-halo terms and a three-haloterm. We distinguish two categoriesof two-halo terms: In
“two-halo-1”,the two shearsignalsare associatedwith the same halo,whilein “two-halo-2”the shearsignalsoriginatefromthe
lenshaloandadifferentneighbouringhalo.Inanalogytothecalculationsforthecorrelator ,onlytermsassociatedwiththeleading
orderclustering correlationfunctionsin the halo correlators ... (termswith ω) and .G.. (terms with Ω) are of interest for
i,j i,j,k
h i h i
theconnectedterms;for ... alltermsareconnected.Goingthroughtheaveragesstepbystepandcollectingtheconnectedterms,
i
yieldsasfinalresultforθh ,iθ ,θ :
1 2 3
G (ϑ ,ϑ ,φ )=G1h(ϑ ,ϑ ,φ )+G2h1(ϑ ,ϑ ,φ )+G2h2(ϑ ,ϑ ,φ )+G3h(ϑ ,ϑ ,φ ). (31)
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
± ± ± ± ±
Westartwiththethree-haloterm,whichisafterperformingtheintegralvariabletransformationsθ θ+θ andθ θ +θ :
3 ′ ′ 3
7→ 7→
G3±h(ϑ1,ϑ2,φ3) := n2ge−2i(ϕ1∓ϕ2)Z d2θd2θ′Ω(cid:16)|θ−θ3|,|θ′−θ3|,|θ−θ′|(cid:17)γh(θ1−θ)γ±h(θ2−θ′) (32)
= n2ge−2i(ϕ1∓ϕ2)Z d2θd2θ′Ω |θ|,|θ′|,|θ′−θ| γh(θ13+θ)γ±h(θ23+θ′).
(cid:16) (cid:17)
Tocastthisintoaformthatnolongerexplicitlycontainsanyϕ,weneedtodoafewmoretransformations.Wefirstnotethatfora
i
shiftedtangentialshearonehas:
θ+θ θ
γh(θ+θ′)=−(θ+θ′′)∗ γt,h(|θ+θ′|)=−e+2iϕ′∆(cid:18)θ′,ϕ−ϕ′(cid:19) γt,h(cid:16)Υ(θ,θ′,ϕ−ϕ′)(cid:17), (33)
8
Simonetal.:TowardsanunderstandingofG3L
where
se+iφ+1 1+2scosφ+s2cos(2φ) i2s(1+scosφ)sinφ
∆(s,φ):= se iφ+1 = 1+s2+2−scosφ ; ∆∗(s,φ)=∆−1(s,φ) (34)
−
isanadditionalphasefactor;ϕandϕ arethepolaranglesofθandθ ,respectively;ΥisgivenbythepreviousEq.(24).Thenthis
′ ′
allowsustorewritethepreviousequationforG3has
±
G3h(ϑ ,ϑ ,φ )
1 2 3
:±= n2g✘e−2✘i(ϕ✘1∓ϕ✘2)Z d2θd2θ′Ω(cid:16)|θ|,|θ′|,|θ′−θ|(cid:17)✟e2i✟ϕ1∆ ϑθ1,ϕ−ϕ1!γt,h(|θ13+θ|)✟e∓✟2i✟ϕ2∆± ϑθ′2,ϕ′−ϕ2!γt,h(|θ23+θ′|)
θ θ
= n2 dθθdθθ dϕdϕ∆ ,ϕ ϕ ∆ ,ϕ ϕ Ω θ,θ ,Υ(θ,θ ,ϕ ϕ) γ Υ(ϑ ,θ,ϕ ϕ ) γ Υ(ϑ ,θ,ϕ ϕ )
gZ ′ ′ ′ ϑ1 − 1! ± ϑ2 ′− 2! (cid:16) ′ ′ ′− (cid:17) t,h(cid:16) 1 − 1 (cid:17) t,h(cid:16) 2 ′ ′− 2 (cid:17)
2π 2π θ θ
= n2 ∞dθθ ∞dθθ dϕ dϕ∆ ,ϕ ∆ ,ϕ Ω θ,θ ,Υ(θ,θ,ϕ ϕ+φ ) γ Υ(ϑ ,θ,ϕ) γ Υ(ϑ ,θ ,ϕ) . (35)
gZ0 Z0 ′ ′Z0 Z0 ′ ϑ1 ! ± ϑ2 ′! (cid:16) ′ ′ ′− 3 (cid:17) t,h(cid:16) 1 (cid:17) t,h(cid:16) 2 ′ ′ (cid:17)
The(connected)two-halotermscanbesplitintotwosub-groups.Onegroupisinsensitivetoshapevariationsofthelenshalo,
asitonlycontainsthemeanhaloshearprofileγ ,
h
G2±h2(ϑ1,ϑ2,φ3) := nge−2i(ϕ1∓ϕ2)Z d2θω(cid:16)|θ3−θ|(cid:17)(cid:16)γh(θ1−θ3)γ±h(θ2−θ)+γh(θ1−θ)γ±h(θ2−θ3)(cid:17)
= nge−2i(ϕ1∓ϕ2) d2θω(θ) γh(θ13)γ±h(θ23+θ)+γ±h(θ23)γh(θ13+θ)
Z
(cid:16) (cid:17)
2π
= n¯ γ (ϑ ) ∞dθθγ (θ) dϕcos(2ϕ)ω Υ(ϑ ,θ,ϕ)
g t,h 1 t,h 2
Z Z
0 0 (cid:16) (cid:17)
2π
+ n¯ γ (ϑ ) ∞dθθγ (θ) dϕcos(2ϕ)ω Υ(ϑ ,θ,ϕ)
g t,h 2 t,h 1
Z Z
0 0 (cid:16) (cid:17)
= γ (ϑ ) γ(ϑ ) γ (ϑ ) +γ (ϑ ) γ(ϑ ) γ (ϑ ) . (36)
t,h 1 t 2 − t,h 2 t,h 2 t 1 − t,h 1
(cid:16) (cid:17) (cid:16) (cid:17)
The last step exploits the two-halo term of our previous result (12) for GGL. The terms inside the brackets express the excess
tangentialshearduetohaloesclusteringaroundthelenshalo.ThestepsinthecalculationofG2h2 arebyandlargeidenticaltothe
stepsundertakeninSect.2.2. ±
Theremainingtermsin(31)arespecial,astheyareindeedsensitivetohalovariations,whichsetsthemclearlyapartfromall
aforementionedcorrelationfunctions.The(connected)one-halotermis
G1±h(ϑ1,ϑ2,φ3):=e−2i(ϕ1∓ϕ2)(cid:28)γh(θ1−θ3;α)γh±(θ2−θ3;α)(cid:29)α =e−2i(ϕ1∓ϕ2)(cid:28)γh(θ13;α)γh±(θ23;α)(cid:29)α . (37)
Ifwewritethehaloshearassumofthemeanshearprofileandsomefluctuationδγ (θ;α)aboutit,i.e.,
h
γ (θ;α)=γ (θ)+δγ (θ;α), (38)
h h h
with δγh(θ;α) α =0,thentheone-halotermbecomes
h i
G1h(ϑ1,ϑ2,φ3) = e−2i(ϕ1∓ϕ2) γh(θ13)γ±h(θ23)+ δγh(θ13;α)δγh±(θ23;α)
± (cid:18) (cid:28) (cid:29)α(cid:19)
= γt,h(ϑ1)γt,h(ϑ2)+e−2i(ϕ1∓ϕ2) δγh(θ13;α)δγh±(θ23;α)
(cid:28) (cid:29)α
= γ (ϑ )γ (ϑ )+ δγ (ϑ ,ϕ ;α)δγ (ϑ ,ϕ ;α) . (39)
t,h 1 t,h 2 t,h 1 1 t±,h 2 2
(cid:28) (cid:29)α
Inthelastequation,weemployed
γ (θ,ϕ;α):= e 2iϕγ (θ;α); γ (θ)= γ (θ,ϕ;α) , (40)
t,h − − h t,h (cid:28) t,h (cid:29)α
whereϕisthepolarangleofθ;anequivalentdefinitionisemployedforthefluctuationsδγ .Owingtostatisticalisotropyofthe
t,h
shearfield,themodelhaloshearprofileγ (θ;α)hastohavearandomorientation.Therefore,thecorrelatorinthepreviousequation
h
mustbeinvariantwithrespecttoanyrotationφ,or
1 2π
δγ (ϑ ,ϕ ;α)δγ (ϑ ,ϕ ;α) = dφ δγ (ϑ ,ϕ +φ;α)δγ (ϑ ,ϕ +φ;α)
(cid:28) t,h 1 1 t±,h 2 2 (cid:29)α 2πZ0 (cid:28) t,h 1 1 t±,h 2 2 (cid:29)α
1 2π
= dφ δγ (ϑ ,φ+φ ;α)δγ (ϑ ,φ;α)
2πZ0 (cid:28) t,h 1 3 t±,h 2 (cid:29)α
=: γ (ϑ )γ (ϑ )δG1h(ϑ ,ϑ ,φ ). (41)
t,h 1 t,h 2 1 2 3
±
9
Simonetal.:TowardsanunderstandingofG3L
Theremainingtwo-halotermin(31)issimilartotheone-haloterm,actuallyanintegraloverG1h:
±
G2±h1(ϑ1,ϑ2,φ3) := nge−2i(ϕ1∓ϕ2)Z d2θω(cid:16)|θ−θ3|(cid:17)(cid:28)γh(θ1−θ;α)γh±(θ2−θ;α)(cid:29)α
= nge−2i(ϕ1∓ϕ2)Z d2θω(θ)(cid:28)γh(θ13+θ;α)γh±(θ23+θ;α)(cid:29)α
= ng✘e−2✘i(ϕ✘1∓ϕ✘2)Z d2θω(θ)✟e+2✟i✟ϕ1✟e∓2✟i✟ϕ2∆ ϑθ1,ϕ−ϕ1!∆± ϑθ2,ϕ−ϕ2!G1±h(cid:16)Υ(ϑ1,θ,ϕ−ϕ1),Υ(ϑ2,θ,ϕ−ϕ2),ν(cid:17)
2π θ θ
= n ∞dθθ dϕω(θ)∆ ,ϕ+φ ∆ ,ϕ G1h Υ(ϑ ,θ,ϕ+φ ),Υ(ϑ ,θ,ϕ),µ , (42)
gZ0 Z0 ϑ1 3! ± ϑ2 ! ± (cid:16) 1 3 2 (cid:17)
wheretheangleνspannedbyθ +θandθ +θandthecorrespondingangleµforaθrotatedbyϕ areimplicitlygivenby
23 13 2
eiν = θ23+θ|θ13+θ| = ϑ2e−i(ϕ−ϕ2)+θ Υ(ϑ1,θ,ϕ−ϕ1) ; eiµ = ϑ2e−iϕ+θ Υ(ϑ1,θ,ϕ+φ3) . (43)
θ13+θ|θ23+θ| ϑ1e−i(ϕ−ϕ1)+θ Υ(ϑ2,θ,ϕ−ϕ2) ϑ1e−i(ϕ+φ3)+θ Υ(ϑ2,θ,ϕ)
4.2.Interpretation
The resulting G is the lowest-order galaxy-galaxy lensing correlation function that, at least within the framework of the ILM,
is sensitive to va±riationsamong shear profiles of haloes. Nevertheless, we also have a G signal when all halo shear profiles are
identical, i.e., δγ = 0, which is generatedby the tangential shear of the lens halo and±the excess tangentialshear of clustering
t,h
neighbouringhaloes.Only(39)and(42),whichbothcontainthecorrelatorδG1h,areaffectedbyascatterinhalomatterprofiles.The
one-halotermofG remainsunchanged,evenifweallowforgeneralcorrelationsbetweenhaloparametersαofdistincthaloesor
forcorrelationsbetw±eenlenspositionsandα,sinceastatisticalindependenceofhaloeshasnotbeenusedforthisterm.Therefore,
weexpectthebehaviourofG onsmallangularscalestobedescribedgenerallybyG1h,notjustwithintheILM.
Inthesimplestcasethata±llhaloshearprofilesareexactlyidentical,γ (θ;α)=γ (±θ),wefind
h h
G1h(ϑ ,ϑ ,φ )=γ (ϑ )γ (ϑ ); δG1h(ϑ ,ϑ ,φ )=0, (44)
1 2 3 t,h 1 t,h 2 1 2 3
±
i.e.,theone-halotermhasnoexplicitdependenceontheopeningangleφ .NotethatG1h andG1h areidentical.Asillustrationof
3 +
the impactof variancein halo shear profiles,consider a singularisothermalellipsoid (SIE)profi−le with ellipticity ǫ and random
h
orientationφ(Mandelbaumetal.2006a)
e+2iϕ ǫ
γsie(θ) 1+ h cos(2ϕ+2φ) , (45)
h ∝− θ (cid:18) 2 (cid:19)
ϕisthepolarangleofθ.Theabsoluteamplitudeoftheshearprofileisnotofinteresthere.ThetangentialshearprofileoftheSIEis
1 ǫ
γsie(θ,ϕ) 1+ h cos(2ϕ+2φ) . (46)
t,h ∝ θ (cid:18) 2 (cid:19)
Inthiscase,wefindbymarginalisingoverallorientationsφ:
1 2π ǫ2 1
G1h(ϑ ,ϑ ,φ )= dφγsie(ϑ ,φ ) γsie(ϑ ,0) ± γ (ϑ )γ (ϑ ) 1+ h cos(2φ ) ; γ (ϑ) , (47)
th±ussi1mil2art3othe2pπreZv0iousrestu,hltb1utn3owh wt,hith2anaidd∝itiont,halφ13-dt,ehpen2dentter8m,or 3  t,h ∝ ϑ
ǫ2
δG1h(ϑ ,ϑ ,φ )= h cos(2φ ). (48)
1 2 3 3
± 8
Theresultbecomessomewhatmorecomplicatedforgeneralslopesδ(AppendixC) andwillrevealadifferencebetweenG1h and
+
G1h whenδ , 1 (notSIE) and ǫ , 0 (elliptical).Moreover,the correlator(41)will exhibitno φ -dependence,if the lenshaloes
h 3
ar−ealwaysaxiallysymmetric,eventhoughtheirradialmatterdensityprofileortheirmassmayscatterastobeexpectedinreality.
Therefore,weconcludethatG mayinprinciplebeusedtoconstraintheshapeor,morespecifically,themeansecond-momentof
theprojectedhalomatterdensi±typrofiles.Inadditiontothat,fluctuationsδγ inthehaloshearprofileduetohalosubstructurealso
t,h
addtothevariancedependentone-andtwo-halotermofG .Aswith theforegoing , itmaybeusefultodefineanexcess∆G ,
obtainedbysubtractingofftheG -signalasanticipatedfrom± theILMwithparameteGrsfromlensclusteringandGGL.IntheILM±
regime,theexcesssignal∆G exa±ctlyvanishes,ifthereisnoscatterinthe(projected)matterdensityprofiles.
±
5. Conclusions
InordertogainabetterunderstandingofG3L,weconceivedatoymodel,the“isolatedlensmodel”(ILM),forthedistributionof
galaxiesandmatteraboutgalaxies.Inthispicture,“isolated”galaxiesaresurroundedbytheirownmatterenvelope(halo).Variations
in the halo matter densityprofile are explicitly allowed,albeit statistically independentto variationsof other matter envelopesor
to positions of other lenses. Consequently, the matter environment of clusters is herein the superposition of independent haloes
producedbyclusteringgalaxies.Theaverageindependentmatterhaloisdescribedbythemeantangentialsheararoundlensesand
theclusteringofthelenses(GGL),Eq.(12).TheforegoingcalculationsevaluatewhatwouldbemeasuredbyG3L(Eqs.20and31)
undertheILMassumptionsanddiscusstheresults.Herewesummariseourmainconclusions.
10