Gravitational lensing: effects of cosmology and of lens and source profiles

13 Pages·0.48 MB·English

by  Francesca Perrotta

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed1February2008 (MNLATEXstylefilev1.4)
Gravitational lensing: effects of cosmology and of lens and source
profiles
F. Perrotta1,2, C. Baccigalupi2, M. Bartelmann3, G. De Zotti1, G. L. Granato1
1Oss.Astr.Padova,Vicolodell’Osservatorio5,I-35122Padova,Italy.Email:[email protected], [email protected], [email protected]
2SISSA/ISAS,AstrophysicsSector,ViaBeirut,4,I-34014Trieste,Italy.Email:[email protected]
3Max-PlanckInstitutfu¨rAstrophysik,P.O.Box1317,D–85741Garching,Germany.Email:[email protected]
1
0 1February2008
0
2

n ABSTRACT
a
J We present detailed calculationsof the magnificationdistribution, includingboth weak

4 andstronglensing,usingveryrecentsolutionsoftheDyer-Roeder(1973)equationforlight
2 propagationinainhomogeneousuniversewithacosmologicalconstantandup-to-datemod-
elsforthe evolvingcosmologicaldistributionofdarkmatterhalos.We addressinparticular

1 theeffectonthemagnificationdistributionofanon-zerocosmologicalconstant,ofdifferent
v densityprofilesoflenses,andoffinitesizesoflensedsources,threeimportantissuesnotyet
0
fullysettled.
3
We showthat,ifdarkmatterfluctuationsarenormalizedtothelocalclusterabundance,
4
inthepresenceofacosmologicalconstanttheopticaldepthforlensingdecreasescompared
1
to the case of an Einstein-de Sitter universe, because halos in the relevant mass range are
0
1 lessabundantoveralargeredshiftinterval.Wealsodiscussthedifferencesinthemagnifica-
0 tion probabilitydistributionsproducedby Navarro,Frenk & White (NFW) and by Singular
/ IsothermalSphere(SIS)densityprofilesoflenses.WefindthattheNFWlensismoreefficient
h
formoderatemagnifications(2 < A < 4), and less efficientforlargermagnifications.More-
p
∼ ∼
- over,wediscussquantitativelythemaximummagnification,Amax,thatcanbeachievedinthe
o caseofextendedsources(galaxies)withrealisticluminosityprofiles,takingintoaccountthe
r
t possibleellipticityofthelenspotential.WefindthatplausiblevaluesofAmaxareintherange
s 10–30.
a
Finally,weapplyourresultstoaclassofsourcesfollowingtheluminosityevolutiontypical
:
v foraunifiedschemeofQSOformation.Wefindthatthelensedsourcecountsat850µmcan
Xi belargerthantheunlensedonesbyseveralordersofmagnitudeatfluxdensities>100mJy.
∼
r 1 INTRODUCTION profiles,onthetheluminosityfunctionandsizeofthesources,and
a
onacorrecttreatmentofthedistance-redshiftrelations,takinginto
Gravitationallensingiswellknowntobeapowerfultooltoprobe accounttheeffectofdensityinhomogeneitiesonthepropagationof
theoverallgeometryoftheuniverseatz<6andthevaluesofcos- lightrays.
∼
mologicalparameters(curvature,vacuumenergydensitydescribed Gravitationallensingbyinterveningmassconcentrationsisa
eitherbyacosmologicalconstantorbyadynamicalquantitylike statisticalprocess.Itcanbedescribedbyaprobabilitydistribution
quintessence, Hubble constant: Bartelmann et al. 1997; Falco et F(A,zs)ofthemagnificationfactorA,whichobviouslydependson
al.1998; Cooray 1999; Huterer &Cooray1999; Macias-Perez et thesourceredshiftzs.ItisgenerallydifficulttomodelF(A,zs)in
al. 2000; Bhatia 2000; Helbig 2000), the evolution of large scale detailanalytically.SomeauthorsestimateditwithN-bodysimula-
structure (Rix et al. 1994; Mao & Kochanek 1994; Bacon et al. tions (see e.g. Rauch 1991; Pei 1993a,b). Nonetheless, the main
2000),themassesandthedensityprofilesofdarkhalosofgalaxies features of F(A,zs) can be captured using generic properties of
andgalaxyclusters(Narayan1998andreferencestherein;Mellier gravitational lensing. Strong lensing occurs near caustic curves,
1999;Cloweetal.2000). which cause F(A,zs) to decrease (cid:181) A−3 for A 1, irrespective
≫
Anotherimportanteffectofgravitationallensingisthemodi- ofthelensmodel(Peacock1982;Turneretal.1984).
ficationoftheobservedluminosityfunctionsofdistantsourcesand Inthispaperwepresentdetailedcalculationsofthemagnifi-
of their counts, due to the redshift dependent magnification bias cationdistribution,includingbothweakandstronglensing,using
(Peacock1982;Narayan1989;Schneider1992). veryrecentsolutionsoftheDyer-Roeder(1973)equationforlight
Inviewofalltheaboveapplications,itisessentialtoperform propagationinainhomogeneousuniversewithacosmologicalcon-
themostaccuratemodelingofalltheingredientsenteringintothe stant(Demianskietal.2000)tocomputetherelationshipsbetween
gravitational lensing process. Quantitative predictions depend on distancesandredshiftaswellasup-to-datemodelsfortheevolving
thelenspopulationasafunctionofmassandredshift,onthelens cosmological distribution of dark matter halos (Sheth & Tormen
(cid:13)c 0000RAS
2 PerrottaF.,BaccigalupiC., BartelmannM., DeZottiG., GranatoG.L.
1999;Shethetal.1999)thatcanactaslenses.Weaddressinpar- A useful expression for s (M) and its derivative ds /dM was ob-
ticular the effect on the magnification distribution of a non-zero tainedbyKitayama&Suto(1996),
cosmological constant,ofdifferentdensityprofilesoflenses,ofa
finite size and of luminosity profiles of lensed sources, three im- s(cid:181) (1+2.208md−0.7668m2d+0.7949m3d)−2/(9d), (4)
portant issues not yet fully settled. The magnification probability G =W 0mhexp[ W 0b(1+√2h/W 0m)] being the shape parameter
distributionhasbeenappliedtothecountsat850µm:theluminos- (Bardeenetal.1−986)andwehaveadoptedapresentbaryondensity
ityfunction of the source population used here has been inferred W 0bh2=0.03.ForaL CDMmodelwithW 0L =0.7andW 0m=0.3,
bytheevolutionoftheQSOsluminosityfunctionandfromthelink theCOBEnormalizationiss 8=s (R=8h−1Mpc)=0.925(Bunn
betweentheQSOsandthehostingspheroids(Granatoetal.2000). &White1997).
SuchluminosityfunctionturnsouttofittheSCUBAdata. AlsoinEq.(1),d 2(z)isthelineardensitycontrastofanobject
c
Theplanofthispaperisasfollows.InSect.2and3,wede- virializingatz,evaluatedatthepresentepoch.Itcanbeestimated
scribe our assumptions on the distribution of lenses in mass and using the spherical collapse model (e.g. Peebles 1980; Lahav et
redshift,andontheir densityprofiles,respectively. InSect.4,we al.1991;Lacey&Cole1993;Nakamura1996;Eke,Cole&Frenk
review the basic definitions of distance and flux magnification in 1996;Łokas&Hoffman2000)yielding:
inhomogeneousuniverses.InSect.5,weintroducetheprobability
forasourceatgivenredshifttobemagnifiedbyacertainamount, d c(z)= d cD+(z=0) , (5)
and quantify the magnification bias on source counts, for which D+(z)
a model is described in Sect. 6. In Sect. 7 we face the problem
whereD+(z)isthelineargrowthfactorofdensityfluctuations:
of determining themaximum magnificationallowed for extended
sourcTesh.roWueghsouumtmthairsizpeaapnedr,dW is0cmusasnoduWrr0eLsudletsnointeStehcet.p8r.esent-day D+(z) ≃ (1+z)−152W m
densityparametersforthenon-relativisticmatterandforthecos- [W 4m/7 W L +(1+W m/2)(1+W L /70)]−1, (6)
mological components, respectively, neglecting the radiation en- × −
Werg0Ry den1sityW .0TmheWH0uLbbislethceocnusrtvanattuirsetHe0rm=,a1n0d0Whkdmenso−t1esMthpec−to1-, izn)3t/hEe2ap(zp)r,oWxiLm=atiWon0Lb/yEC2a(rzr)o,lalnedtal.(1992),whereW m=W 0m(1+
atanlddhe≡n=si0ty−.6p5a.rFamur−ethteerr.mOourrefi,qd0uc=ialma¨ao/dae˙2lhisasthWe0dmec=el0er.3at,iWon0Lpa=ra0m.7- E(z)=H(z)/H0=[W 0m(1+z)3+W 0R(1+z)2+W 0L ]1/2. (7)
eter,andweconcentrateonspati−ally-flatmodels(W 0R =0). d c[Eq.(5)]isthevalueofthecriticaloverdensityevaluatedatviri-
alization. An approximate expression valid for any cosmological
model was reported by Nakamura (1996) and Kitayama & Suto
2 THEEVOLVINGHALOMASSFUNCTION (1996):
Weassumethatthelenspopulationconsistsofcollapseddarkmat- 3(12p )2/3
terhaloswithanepoch-dependent massfunctiondescribedbythe d c≈ 20 (1+0.0123log10W m). (8)
Sheth & Tormen (1999) function, which reproduces fairly accu-
ratelytheresultsofextensivenumericalsimulationsovermorethan
fourordersof magnitudeinmass, forawiderangeof ColdDark
3 GRAVITATIONALLENSINGANDHALOMODELS
Mattercosmologies (Jenkinsetal.2000). Thisfunction improves
considerablyonthefamiliarPress-Schechter(1974) modelwhich Theray-tracingequationrelatesthepositionofasourcetotheim-
overestimatestheabundanceof“typical”(M⋆)halosandunderes- pact parameter in the lens plane of a light ray connecting source
timatesthatofmassivesystems.Thecomovingnumberdensityof andobserver.Thelightraypassingthelensatanimpactparameter
haloswithmassMatredshiftzisthen ~z isbent byanangleaˆ(~z ).Thesourceposition~h andtheimpact
dn = 2aA2 r 0 d c(z) 1+ s (M) 2p parameter~z inthelensplanearerelatedthrough
dM rdlnps M2 s (Ma)d "2(z) (cid:18)√ad c(z)(cid:19) # ~h = DDds~z −Dds~aˆ(~z ), (9)
exp c ; (1)
× (cid:12)dlnM(cid:12) (cid:18)−2s (M)2(cid:19) whereDd,s,ds aretheangular-diameterdistancesbetweenobserver
(cid:12) (cid:12) and lens, observer and source, and lens and source, respectively.
Thebest-fit(cid:12)values(cid:12)of theparametersarea=0.707, p=0.3, and
(cid:12) (cid:12) Wewill specify these distances later inthe context of a “clumpy
A 0.3222(Sheth&Tormen1999; Sheth,Mo&Tormen1999).
≃ universe”.Thedeflectionangle~aˆ isthesumofthedeflectionsdue
ThePress-Schechter massfunction isrecovered for a=1, p=0
toallmasselementsofthelensprojectedonthelensplane,
andA=0.5.
InEq.(1),r 0 isthemeanmassdensityatareferenceepoch 4G ~z ~z
to0f,lwinheiacrhdwenesaitsysuflmucetutoatbioentsheatptrheesepnrtestiemnte,epanodchs s2misooththeevdawriaitnhcea ~aˆ(~z )= c2 ZR2 ~z −~z ′2S (~z ′)d2z ′, (10)
sphericaltop-hatfilterWR(k)enclosingamassM.Thevariances 2 (cid:12) − ′(cid:12)
isrelatedtothepowerspectrumP(k)by: whereS isthesu(cid:12)(cid:12)rfacem(cid:12)(cid:12)assdensityofthelens.
Itisconvenienttointroducethedimensionlessimpactparame-
4p ¥
s 2(R)= (2p )3 Z0 dkk2WR2(k)P(k), (2) itnerg~xv=al~uz e/zp0r,owjehceterdezin0itsheansoaurbrciteraprylalneengisthh s0ca=lez.0TDhes/cDordreasnpdonthde-
with sourcepositionis~y=~h /h 0.
3 Thedimensionlesssurfacemassdensityk (x),alsocalledcon-
WR(k)= (kR)3[sin(kR)−(kR)cos(kR)]. (3) vergence,is
(cid:13)c 0000RAS,MNRAS000,000–000
Gravitationallensing 3
k (x)= SS(cxr) with S cr= 4cp 2G DdDDsds . (11) tMheadmauas2s0a0n0d;Ktoaitsheerv1i9r8ia6l)i:zation redshift by (see, e.g. Porciani &
~Tyh=e~xd−im~aen(~xs)io,nlessray-tracingequationisthen, (12) s v= 21H0r0W 10m/3D 1/6(cid:20)WW 0mm(cid:21)1/6(1+z)1/2 (22)
withthescaleddeflectionangle whereW m isdefinedintheprevioussection,r0=(3M/4rp 0)1/3,
D D 2 andD (z)isthemeandensityofthevirializedhaloinunitsofthe
~a (~x)= z d0Ddss~aˆ(z 0~x)= x Z dx′x′k (x′). (13) cdreinticcealodfesnvsiotnyraetdtshhaitftreisdsghivifetn.AinuBsreyfuanle&xpNreosrsmioannf(o1r99th8e),depen-
The axial symmetry of the lenses considered here allows us to
rewritethelensequationintermsofthedimensionless“mass”m(x) s v=M1/3[H2(z)D (z)G2/16]1/6. (23)
enclosedwithintheradiusx,
Bryan&Norman(1998)alsogiveafittingformulaforD (z)fora
m(x)
~y=~x , (14) flatuniversewithacosmologicalconstantandforauniversewith
− x W 0R =0andW L =0.Foraflatuniverse,(W 0R =0)theirhydro-
6
with dynamicalsimulationsyield:
x
m(x)=2 dx′x′k (x′). (15) D (z)=18p 2+82x 39x2, (24)
0 −
Z
TheJacobianmatrixA(~x)=¶ ~y/¶ ~xforthelensmapping[Eq.(14)] wherex=W m 1.
−
determinesthemagnificationfactorforeachimagei, The SIS model is useful because it allows to work out an-
alytically the basic lensing properties. On the other hand, high-
1
µi= detA(~xi) . (16) srehsoowluetdiotnhaNt -inbohdiyerasricmhuiclaatliloyncslu(sNtearvianrgrou,nFivreernskes,&viWriahliitzeed19d9ar7k)
Thesignof µi reflectstheparityoftheimage withrespect tothe matterhaloshaveauniversaldensityprofile(referredtoasNFW),
source. The total source magnification is the sum of theabsolute whichisshallower thanisothermal near thecenter and steeper in
valuesofthemagnificationfactorsofallimages, theouterregions:
µ=(cid:229) |µi|. (17) r (x)= r critd NFW , (25)
i x(1+x)2
Foraxiallysymmetriclenses,theJacobiandeterminantofthelens
mappingcanalsobewritten wherex=r/rs,rs beingascaleradius,d NFW isthecharacteristic
densitycontrast of thehalo, andr isthecriticaldensity at the
crit
detA = 1 a (x) 1 d a (x) epoch of the halo virialization. This formula was found to accu-
− x −dx ratelydescribetheequilibriumdensityprofilesofdarkmatterhalos
(cid:18) (cid:19)(cid:18) (cid:19)
= 1−xm2 1−ddx mx =(1−k )2−g 2, (18) iorvreesrpaecbtriovaedorfatnhgeecoofsmmoalsosgeisca(3lp×ar1a0m11et≤ersMa2n0d0/tMhe⊙in≤iti3al×d1en0s1i5t)y,
(cid:16) (cid:17)(cid:20) (cid:16) (cid:17)(cid:21) fluctuationspectrum.
whereg istheshear(cf.Schneider,Ehlers&Falco 1992).Critical
Stillhigher resolution N-body simulationsindicate asteeper
curvesarelocatedatthezerosofdetA;theirimagesinthesource centralcuspthanthatoftheNFWprofile:r (x)(cid:181) x1.5(1+x)1.5 −1
planeunderthemappingdescribedbyEq.(12)arethecaustics.
(Mooreetal.1999;Ghignaetal.2000).Sincetheslopeofthecen-
A simple model for the mass profile of a lens (cluster or tral density profile in this case [r (r)(cid:181) r 1.5] is(cid:2)intermediate(cid:3)be-
galaxy) is the singular isothermal sphere (SIS; e.g. Binney & −
tweenthoseoftheNFW[r (r)(cid:181) r 1]andoftheSIS[r (r)(cid:181) r 2],
Tremaine1987): − −
thetwocasesconsideredhere(NFWandSIS)willbracketit.
s 2 We parameterize halos by their mass M (defined as the
r (r)= v , (19) 200
2p Gr2 masswithinthevirialradiusr200,theradiusof asphere ofmean
wheres v isthe line-of-sight velocity dispersion. Inthiscase, the interior density 200r crit; see Navarro et al. 1997), since this is
deflectionangleisindependentoftheimpactparameter(cf.Schnei- themasswhosedistributionisgivenbyEq.(1).Unlessotherwise
deretal.1992;Narayan&Bartelmann1997).Oneofthetwocrit- specified, we abbreviate M200 by M. The halo concentration is
icalcurvesdegeneratestoapoint.Anygivensourcehaseitherone c=r200/rs.Itisrelatedtothedensityparameterd NFWby
ortwoimages.Twoimagesappearonlyifthesourceliesinsidethe 200 c3
Einsteinring.TheEinsteinanglecorrespondingtoaSISis d NFW= . (26)
3 [ln(1+c) c/(1+c)]
D −
q E=aˆ ds (20) Thevirialradiusofahaloatredshiftzdependsonthehalomassas
Ds
waˆi=th4p sc22v =1.4′′ 220ks mvs 1 2 . (21) Ar2l0t0er=na1ti.v6e(31ly×+,t1hz0)e−h2al(cid:18)ohca−n1MMbe⊙c(cid:19)ha1r/a3c(cid:20)teWrWizme0(mdz)b(cid:21)y−i1ts/3cihr−cu1lkaprcve.loc(2it7y),
(cid:18) − (cid:19)
Inordertoevaluatethemassofahalowithvelocitydispersions v GM 1/2
fromEq.(19), wetruncatethesphere ataneffective radiuscom- V200 = r 200
putedfollowingLahavetal.(1991). (cid:18) 200 (cid:19)
If everyhalo virializestoformasingular isothermal sphere, r W 1/2
massconservationimpliesthatthevelocitydispersionisrelatedto = (cid:18)h−120k0pc(cid:19)(cid:20)W m0(mz)(cid:21) (1+z)3/2kms−1. (28)
(cid:13)c 0000RAS,MNRAS000,000–000
4 PerrottaF.,BaccigalupiC., BartelmannM., DeZottiG., GranatoG.L.
The scale radius rs depends on the halo mass. The halo concen- (e.g.Schneider1984,1987a,b).Thisisthe“emptybeamapproach”
trationincreaseswithdecreasinghalomass(e.g.Fig.6ofNavarro to light propagation (Dyer & Roeder 1973; Ehlers & Schneider
et al. 1997). Lessmassive halosare thereforemore concentrated. 1986),inwhichlightconesaredevoidofclumpedmatter.Thefrac-
Foragivenhalo mass,Eqs. (27) and(26) completely specifythe tionofuniformlydistributedmatterisdenotedbya s,thesmooth-
densityprofile[Eq.(25)]. nessparameter.
Thelensequations fortheNFWprofilearegiven byBartel- In this on-average homogeneous and isotropic universe, the
mann(1996)andMaozetal.(1997).Thesurfacemass-densityis averageflux S fromasourcepopulationatredshiftzwithlumi-
h i
nosityL,mustbeequaltothecorrespondingfluxS thatwouldbe
S (x)= 2r critd NFWrs f(x), (29) observedinastrictlyuniformFriedman-LemaˆıtreuFLniverse,
x2 1
−
with LK(L,z)
hSi=SFL= 4p D¯2(z) , (35)
f(x)= 101−−√√x1222−−x12aarrccttaannhqq((((xx1x−++−111x)))) (((xxx=><1)11)) . (30) Kwmh(eLearn,ez)flD¯iusLx(tehzs)eoiKsf-tshcooeLurlrruecmcetsiinoionns.itIthnyedgseiesntteawrnaocle,diiitnffiteshrneeonuttnpsiofposarscmiebtliuemntieovsec.roIsmne,ppaaanrrde-
Thedimensionlesslensmassis ticular,thenotionofdistancehasnouniquemeaninginaclumpy
universe, asit depends on both redshift and direction. Duetothe
m(x)= 4r critd NFWrsg(x), (31) corrugated structure of the gravitational field, the propagation of
S
cr light in the inhomogeneous universe is a statistical problem. We
with cancomparefluxesasinEq.(35)onlybecauseweassumethatthe
x global geometry oftheclumpy universe equalsthat ofthehomo-
g(x)=ln +1 f(x). (32)
2 − geneousuniverse.Equation(35)impliesthattheareaofasurface
of constant redshift on the future light cone of a source is equal
WhiletheNFWprofile[Eq.(25)]hastwocriticalcurves(Bartel-
to that of the corresponding wavefront in the Friedman-Lemaˆıtre
mann1996),theSISprofilehasonlyone.ASISlenshaseitherone
model.Inordertouseredshift-distancerelationsinaninhomoge-
ortwoimages,anNFWlenseitheroneorthree.
neous, partlyclumpy universe, which isuniformonaverage, itis
thennecessarytointroduceameaningfuloperationaldefinitionof
distance.Sincelightpropagationinaclumpyuniversedependson
4 THEMAGNIFICATIONBIAS theclumpsinandnearthelightbeam,Dyer&Roeder(1973)con-
sideredthelimitingcaseofalightbundlethatpropagatesfaraway
Inthissection,wefollowtheapproachbySchneideretal.(1992) fromallclumpsandisthusunaffectedbygravitationalshear.This
andSchneider (1987a,b).Wecallµthemagnificationoffluxesin isthe“emptycone”limit.
aninhomogeneousuniverse,andrelateittothemagnificationAin Theangular-diameterdistanceisthencomputedbyreplacing
ahomogeneousuniverse. thedensityparameterW mwiththe“homogeneous”fractiona sW m.
Theeffectoflensingonflux-limitedsourcecountsisquanti- The resulting “Dyer-Roeder” distance is larger than the angular-
fied by the magnification bias (Turner et al. 1984). If the surface diameter distance in ahomogeneous universe. For a spatially-flat
densityofgalaxieswithfluxgreaterthanSn isN(Sn ),fluxmagni- universe with deceleration parameter q0 =0.5, the Dyer-Roeder
ficationaltersthecountstoN′(Sn )=µ−1N(Sn µ−1),andthemag- distancebetweenredshiftsz1andz2>z1is
nificationbiasisgivenby(Narayan1989):
q(µ,Sn )= NN′ = Nµ(NSn(µS−n )1) . (33) r(z1,z2,b )= b2 "((11++zz21))((bb −+55))//44−((11++zz21))((bb +−55))//44#, (36)
The factor µ−1 arises because solid angles are magnified, hence where b =(1+24W G)1/2 and W G =(1 a s)W m is the density
source counts are diluted. Eq. (33) implies that for power-law parameter in compact objects (b =a s =−1 for the homogeneous
counts(N(cid:181) Sn−a ),q(µ,Sn )=µa −1,sothatthemagnificationbias universe).Hereafter,weabbreviater(0,z,b ) r(z,b )andr(z,1)
≡ ≡
increasesasthecountssteepen. r1(z). The dimensional angular diameter distance in a homoge-
Sincethereferenceto“unlensed”countshasgivenrisetocon- neousuniverseis
fusion in the literature, it is important to specify our model for c
light propagation in the universe when talking about the number D1(z)= H r1(z). (37)
0
ofsourcesseen“inabsenceoflensing”.
InEq.(33),Nisdefinedasthenumbercountofsourcesthatdo Theproblemismorecomplicatedwhenthecosmologicalconstant
notappearbehindlenses.Becauseofenergyconservation,theflux ispositive.TheDyer-RoederequationinaFRWuniversewithnon-
oftheseobjectsmustbereducedcomparedtoahomogeneousuni- zerocosmologicalconstanthasbeensolvedexactlybyKantowski
versewiththesameaveragedensity.Inotherwords,sincelensing (1998), Kantowski & Kao (2000) and, following a different ap-
iscausedbyinhomogeneities,wehavetodealwithaninhomoge- proach, by Demianski et al. (2000). The general solution can be
neousuniverse.Whilesourcesobservedthroughlensesaremagni- expressedintermsofthehypergeometricfunctions fs introduced
fiedbyafactorµ+>1comparedtoahomogeneousuniverse,un- byDemianskietal.(2000): ±
lensedsourcesmustbedemagnifiedbyµ <1.Whencomparing
lensedtounlensedsources,theeffectivem−agnificationis r(z,b )=A1((11++zz))−5b//44 (1f+s+z)3+A2((11++zz))b5//44 (1f+s−z)3 , (38)
µ+
µ= >1 (34)
µ− wheretheconstantsA1,2aredeterminedbytheboundaryconditions
(cid:13)c 0000RAS,MNRAS000,000–000
Gravitationallensing 5
r(z,b ) =0 and dr =1. (39) Thisisthedefinitionofmagnificationused,forexample,byBlain
|z=0 dz (1996) and Peacock (1982). It is trivial to show that for a given
(cid:12)z=0
(cid:12) sourceredshiftz, p(µ,z)dµ=p(A,z)dA,andthenormalizationand
Thiscanbeusedtogeneral(cid:12)izethetwo-pointdistanceofEq.(36),
(cid:12) fluxconservationconditionsbecome
which,foranon-zerocosmologicalconstant,reads:
¥ ¥
r(z1,z2,b ) = r(z1,b )(1+z1)r(z2,b ) ZAmindAp(A,z)=1, ZAmindAp(A,z)A=1. (48)
× Zz1z2dz′[(1+r2z(′z)′3,Wb )0(m1++Wz′0)L2]1/2 , (40) Hveerrsee,,Athmaitncisanthbeempirnoidmuucmedmbyagcnoimficpaatciotnlerneslaetsi,vie.et.oAthmeinF=RWhµiu−n1i-.
withr(z,b )givenbyEq.(38). SincehAi=1,thesourceswhicharedemagnifiedwithrespectto
thehomogeneousuniversehaveA<1.
For practical applications, these exact solutions are rather
cumbersometouse.Demianskietal.(2000)giveapproximateso-
lutionsforr(z,b )andr(z1,z2,b ),whichweuseinthispaperinthe
5 THEMAGNIFICATIONDISTRIBUTION
form kindly provided by R. de Ritis(private communication) for
W 0L =0.7anda s=0.9,implyingb =1.84,andW G=0.1.Weig- Thenumberofimagesproducedbyalens,theirangularseparation
noreapossibletime-dependenceofthesmoothnessparameter.The andtheirmagnificationdependontheirrelativealignmentasseen
effectofdifferentchoicesofa s ontheDyer-Roederdistancewith bytheobserver.Forafixedgeometryofthelenssystem,onecan
positivecosmologicalconstantisshowninDemianskietal.(2000). askwherethesourcemustlieforitsimagestohaveatotalmagni-
Having introduced the Dyer-Roeder distance, we can inter- ficationlargerthanagivenvalueµ.Theareaoftheresultingregion
pret themagnification inEq. (34) as theratio between the flux S inthesourceplaneisthecrosssections (µ,zd,zs,c ),whichobvi-
actually received from a source at redshift z, and the flux Sempty ouslydepends onlensandsourceredshifts, zd andzs,onasetof
thatwouldbereceivedifthesamesourcewasobservedthroughan parametersc describingthelensmodel,andonthemagnification
emptycone: itself. As both SIS and NFW halos are completely characterized
S 4p D2(z,b ) by their mass, c =M here. The cross section quantifies the effi-
µ= =S L >1, (41) ciencyoftheindividuallensonasource.Itisgenerallyfoundnu-
Sempty LK(L,z) merically,bysolvingthelensequationandfindingtheareainthe
where source plane where a source must lieinorder to produce images
DL(z,b )= Hc0(1+z)2r(z,b ) (42) swinitghualartoitsaolthmearmgnailfiscpahtieornefloarrgaertotthaalnmµatgont.ifiTchaetiocrnoµsstotse>ctµioinsof a
istheluminositydistanceinanemptycone. 4p aˆ2D2
Since the clumpy universe is homogeneous on-average, s (µtot>µ)= µ2 ds for µ≥2. (49)
D¯L(z,b )=DL(z,1).UsingEq.(41),Eq.(35)reads We need to compute the total magnification cross section of an
r(z,b ) 2 ensemble of lenses distributed according to the mass function of
hµi=(cid:18) r1(z) (cid:19) >1. (43) Eanqd.(n1()z.,WMe)den(o1te+bzy)3nncc((zz,,MM))t.hAescolmonogviansgtlheenscrnousmsbseerctdioennssitoyf,
µ canbeinterpretedasthemagnificationofanaveragelightbeam individual le≡nsesdo not overlap, eachlight bundle fromasource
h i
inasmoothuniverserelativetothatforanemptyconeinaclumpy encountersonlyonelens,andthetotalcrosssectionistheintegral
universe: µ =SFL/Sempty. oftheindividualcrosssectionsovertheredshiftandmassdistribu-
h i
Forq0=1/2, µ becomes tionsoflenses:
h i
sinh2[0.25b ln(1+z)] c 3
hµi=b −2 sinh2[0.25ln(1+z)] , (44) s tot(µ,zs) = 4p (cid:18)H0(cid:19) (50)
(Pei1993b),whileforthemoregeneralcaseincludingthecosmo- zsdz dMs (µ,z,zs,M)nc(z,M)(1+z)2r12(z) ,
logicalconstant,hµihastobederivedfromEq.(43). × Z0 Z W 0m(1+z)3+W 0L
Let p(µ,z)dµdenotetheprobabilityforasourceatredshiftz
wherethepropervolumeofasphericaplshellofwidthdzatredshift
tobemagnifiedbyafactorofµwithindµ.Thenormalizationand
zis
fluxconservationconditionsrequire:
¥ ¥ dV =4p cr1(z) 2 drpropdz. (51)
Z1 dµp(µ,z)=1, Z1 dµp(µ,z)µ=hµi. (45) (cid:18) H0 (cid:19) × dz
Foraspatially-flatuniverse
Whendealingwithsourcecountsoncosmologicalscales,however,
itisoftenconvenient torefer magnificationstothehomogeneous drprop c
= . (52)
universe rather than to the “demagnified” background. Wecall A dz H0(1+z) W 0m(1+z)3+W 0L
themagnificationrelativetothehomogeneousuniverse:
Theprobabilityforapsourceatredshiftzstobelensedwithmagni-
A= S = Semptyµ, (46) fication>µisobtainedbydividings tot bytheareaofthesource
S S sphere
FL FL
wanheemrepµty=beSa/mSe,mapstyinisEtqh.e(4m1a).gEniqfisc.a(t3i5o)naonfda(4s3o)uricmepslyee:nthrough P(µ,zs) = Hc0 r12(1zs) Z0zsdz W r01m2((z1)+(1z+)3z+)2W 0L
µ
A= µ . (47) · dMs (µ,z,zs,M)pnc(z,M). (53)
h i Z
(cid:13)c 0000RAS,MNRAS000,000–000
6 PerrottaF.,BaccigalupiC., BartelmannM., DeZottiG., GranatoG.L.
Therequirementofnon-overlappingcrosssectionsrestrictstheva- wheretheluminositydistanceD2(z,1)iscomputedfromEq.(42)
L
lidityofEq.(53)toP 1,i.e.thetotalcrosssectionmustbemuch withb =1and
≪
smallerthantheareaofthesourcesphere.Theneteffectofgravi-
L(n )
tationallensingonthedistributionoffluxdensitiesexpectedfrom Lmin(n )=4p (1+z)r2(z)Sn L[(1+z)n ] . (57)
apopulationofdistantsourcescanbedescribedbytheprobability
distributionofmagnifications, p(µ,z)introducedinSect.4.While Notethatthecomoving radialcoordinater [anditselementdr in
P 1,thedifferentialprobabilityisp(µ,z)= dP(µ,z)/dµ;inpar- Eq.(56)]mustnotbeconfusedwiththedimensionlessangulardi-
≪ −
ticular,p(A,z)=p(µ,z) µ . ameterdistancer1(z)[Eq.(37))].
Thedifferentialprohbaibilitydecreasesasµ 3forµ 1,hence The luminosity function modified by the magnification bias
−
thehighmagnificationtailintermsofAcanbewrittena≫sp(A,z)(cid:181) reads(e.g.Pei1995):
a(z)A−3. On the other hand, Eq. (53) breaks down for small ¥ p(A,z) L
magnifications, where multiple lensing events become important F ′(L,z)= dA F ,z . (58)
and crosssections overlap (infact,the probability for manylow- ZAmin A (cid:18)A (cid:19)
magnificationlensingeventsalongthelineof sighttoasourceis Allowancefortheeffectoflensingoncountsismadebyreplacing
rather large, whilea singleinteractionproducing high magnifica- F (L,z)toF (L,z)inEq.(56).
′
tionsisarelativelyrareevent).Inparticular,thisimpliesthatthere
is a critical magnification Acut below which multiple lensing be-
comesimportant,resultinginlow-magnificationevents(weaklens-
6 SOURCECOUNTSINTHESUBMILLIMETRE
ingregime).Vietri&Ostriker(1983)analyticallydescribedmulti-
WAVEBAND
plelensingfortheSISmodel.
Based on general considerations (see Bartelmann &Schnei- In this paper, the “unlensed” galaxy counts in the submillime-
der 2001 for a review), one expects a skewed magnification dis- tre waveband are taken from the model presented by Granato et
tribution with a weak lensing peak near A=1, turning into the al.(2000),whichisingoodagreementwiththeavailableSCUBA
high-magnificationtail(cid:181) A−3atA=Acut. dataat850µm(Blainetal.1999;Smailetal.1999).
For a Gaussian density fluctuation field d , weak lensing by Inthismodeltherateofformationofspheroidsathighredshift
largescalestructure(e.g.Bartelmann&Schneider2001;Miralda- is estimated exploiting the (i) QSO Luminosity Function and (ii)
Escude´ 1991; Blandford et al. 1991; Kaiser 1992; Jain & Seljak observationalevidenceleadingtotheconclusionthathighredshift
1992) produces a Gaussian magnification distribution. In fact, as QSOsdidshineinthecoreofearlytypeproto–galaxiesduringtheir
longasd A A 1 1,themagnificationofasourceatredshiftz mainepisodeofstarformation.Intheirscenariothestarformation
≡ − ≪
canbeapproximatedas ismorerapidinmoremassiveobjects,rangingfrom 0.5to 2
∼ ∼
Gyrwhengoingfrommoremassivetolessmassiveobjects.This
A(z)=1+d A(z) 1+2k (z), (54)
≈ Anti-hierarchical BaryonicCollapseisexpectedtooccur inDark
i.e.tofirstorderthemagnificationfluctuationisjusttwicethecon- Matter halos, when the processes of cooling and heating is con-
vergence k , which itself is a projection of the density contrast d . sidered. Thelarger thedark haloand theenclosed spheroid mass
Thedistributionofthemagnification fluctuationsd A willthenbe are, theshorter willthegasinfall and cooling timesbe, resulting
Gaussian, withmeanzero andadispersion s A(z)which depends inafasterformationofthestarsandofthecentralblackhole.The
on the source redshift and, albeit weakly, on cosmology. Typical star-formation process and the quasar shining phase proceed un-
valuesfors Arunfrom 2 10−3forz=0.05to 0.44atz=7.5 tilpowerful galacticwindsarecaused by thequasar itself,which
∼ × ∼
(cf.Bartelmann&Schneider2001). occurs at a characteristic timewhen its luminosity becomes high
AconvenientchoiceforAcutisAcut=1+1.5s A(z),yielding enough.AlsoMonacoetal(2000),inordertoaccountfortheob-
Acut 1.5 2 for the redshift range of interest in this paper. We servedstatisticsofQSOsandellipticalgalaxiesintheframeworkof
≈ −
modeltheprobabilitydistributionforA<Acutas hierarchicalstructureformation,introducedaatimedelaydecreas-
ingwithmassbetweenthebeginningofthestarformationandthe
p(A,z)=H(z)exp[−(A−A¯)2/2s 2A(z)], (55) QSObrightphase.
wherethepreciselocationofthepeakA¯andtheamplitudeH(z)are Thespectroscopicevolutionofgalaxiesadoptedhereisbased
on the model GRASIL (Silva et al. 1998). GRASIL includes: (i)
determinedbythenormalizationandfluxconservationconditions
chemical evolution; (ii) dust formation, assumed to follow the
[Eq.(48)]onthecombined(weakplusstronglensing)probability
chemistryofthegas;(iii)integratedspectraofsimplestellarpop-
distribution.
ulations(SSP)withtheappropriatechemicalcomposition;(iv)re-
The magnification probability for isothermal galaxy mod-
alistic 3D distribution of stars, molecular clouds (in which stars
els has been derived by Peacock (1982) and Vietri & Ostriker
formandsubsequentlyescape)anddiffusedust;(v)radiativetrans-
(1983), and, for more complicated galaxy models, by Blandford
fercomputationinthisclumpyISManddusttemperaturedistribu-
&Kochanek(1987)andWallington&Narayan(1993).Inthispa-
tiondeterminedbythelocalradiationfield.
per,weintegrateEq.(53)alsoforlenseswithNFWdensityprofile
Withtheseingredients,theevolvingluminosityfunctions(LF)
(see Sect. 3), and we describe the low-magnification distribution
atvariouswavelengthsinthemillimeterandsubmillimetrewave-
accordingtoEq.(55).NumericalresultsarepresentedinSection8.
bands are evaluated numerically, and they turn out to be signif-
Let us now turn to the magnification bias on a flux-limited
icantly different from models of Pure Luminosity Evolution for
sourcesample.Theintegratedcountsaboveafluxdensitythreshold
Sn ofsourceswithacomovingluminosityfunctionF (L,z)canbe the 60µm LF of IRAS galaxies (Saunders et al. 1990), properly
rescaledtothewavelengthsofinterest.TheIRASgalaxyLF,which
writtenas(seee.g.DeZottietal.1996):
isbasedonanempiricalmodeldescribingtheevolutioninapara-
N(Sn )=Z0z0 dzZL¥mindLF (L,z)D(12L+(zz,)12) ddrz sr−1, (56) msuebtmricillwimaye,trweawsauvseebdabnydsB,lwaihnic(1h9w96e)retothoebntauisnegdafloaxryesctoimunattsinigntthhee
(cid:13)c 0000RAS,MNRAS000,000–000
Gravitationallensing 7
7 MAXIMUMMAGNIFICATIONFOREXTENDED
SOURCES
Asdiscussed byPeacock (1982), inthecase of extended sources
themagnificationcannotbearbitrarilylarge.Correspondinglythe
magnificationdistributionmustbecut-offatlargeAas:
A
p(A)(cid:181) exp , (59)
(cid:18)−Amax(cid:19)
wherethecut-offmagnificationAmaxdependsonthephysicalsize
ofthesource.
Since, in some applications, e.g. for the estimate of the in-
fluence of lensing on counts of sub-millimeter sources (Blain
1996), the results are sensitive to the adopted value of Amax,
theapproximated expression derived by Peacock (1982) [Amax=
70(DH0/c)(d/kpc)−1,wheredisthephysicalradiusofthesource,
D is the angular diameter distance of the source and c/H0 is the
Hubbleradius]maynotbesufficient.
Themorphologyofstronglylensedsourcesindicatesthatmost
lenses are not circularly symmetric (e.g. Narayan & Bartelmann
1997andreferencestherein).Therefore,toestimatethemaximum
Figure1.Magnificationofanextendedsourceasafunctionoftheoffsetr possiblemagnificationforasourceofphysicalradiusratredshift
betweenitscenterandtheprojectionofthelenscenterinthesourceplane. zs, weconsidered in general an elliptical lensing potential due to
Herethelenspotentialhasnoellipticity.Seetextfordetails. aquasiisothermalsphere(Blandford&Kochanek1987).Defining
polar coordinates r and q in the image plane, with origin at the
centerofmass,andmeasuringq fromthemajoraxisoftheellipse,
thedeflectionpotentialmaybewrittenas:
Y (r,q )=q E (s2+r2)(1 gcos2q ). (60)
−
Heregistheqellipticityparameter,sthecoreradiusandq isde-
E
fined by Eq. (20). The results for an extended source are weakly
dependentons,withinarelativelybroadinterval;thereforewesim-
plysets=0inthefollowing.Withthispotentialwehavecomputed
theexpectedmagnificationofapointlikesourceA(~y),asafunction
ofitsposition~yinthesourceplane.Thishasbeendonebymeans
oftheray-shootingmethod(e.g.Schneideretal.1992,pag.304).
Thenthemagnificationofanextendedsourcewithbrightnesspro-
fileI(~y),asafunctionoftheposition~yE ofitscenter,isgivenby
I(~y)A(~y)d2y
AE(~yE)= I(~y)d2y . (61)
R
For the brighRtness profile, we have used either a de Vaucouleurs
law
logI(R)=logIe 3.33 (R/Re)1/4 1 , (62)
− −
h i
Figure2.Contourplotofthemagnificationforanextendedsourcewhen ortheHubbleprofile
gof=ax0e.1s..TThheelpervoejlesctairoenaotfAth=el1e.n5s,2c,e4n,te6r,8in,1t0hefrsoomurcoeutpsliadneeiins.aSteteheteoxrtigfoinr I(R)=Io/(1+R/Ro)2. (63)
details. Observed profiles of spheroidal galaxies are well reproduced by
bothfunctionalformsoverawiderangeofR,providedthatRe
≃
incidenceof gravitationallensingonsourcecounts.However, the 11Ro(Mihalas&Binney1981).
We find that, adopting this relationship between the scale-
modelbyGranatoetal.impliessteepersourcecounts,nearlyexpo-
nentiallydecreasingatbrightfluxes,sothat,asexplainedinsection lengths,theestimatedAE areverysimilarinbothcases.Therefore
inthefollowingwepresentonlyresultsforEq.(62).Weassumein
(4), theeffect of gravitational lensingisexpected tobemore im-
portant. particularRe=5kpc,typicalforabrightellipticalgalaxy(e.g.Kent
1985).ForaSIS,themaximumpossiblemagnificationisachieved
All source counts obtained in this paper, with and without
magnificationbias,thespheroidsincludeellipticalgalaxiesaswell when the lens is local, i.e. when Dds =Ds in Eq. (20). We set
asbulgesofSagalaxies,andwefollowedtheformalismofGranato s v=300kms−1,whichcorrespondstoamassof1012M within
25 kpc adopting the isothermal relationship M(<R)=2⊙s 2R/G.
etal.(2000)onthesourceproperties. v
Withouradoptedcosmology,Eq.(20)yieldsq E=2.6′′.
Figure 1 shows the corresponding magnification for an ex-
tendedsourceasafunction oftheoffsetr between itscenterand
(cid:13)c 0000RAS,MNRAS000,000–000
8 PerrottaF.,BaccigalupiC., BartelmannM., DeZottiG., GranatoG.L.
0.01
1 0.001
0.0001
0.1
0.01
0.001
0.0001
0.01
0.001
0.01
0.001
0.0001
1
Figure3.SISmodel:comparisonofthehigh-magnificationtailamplitudes
a(z)foraflatL CDMmodelwithW L =0.7(solidline),andaflatCDM
model(dottedline),bothnormalizedtotheclusterabundance.Theampli-
tudeisplottedvs.sourceredshift.Thedot-dashedlinereferstoaCOBE-
Figure4.TheSheth&Tormen(1999)massfunctionforaflatCDMmodel
normalizedflatCDMuniverse.
(dotted line) andforaL CDMmodel(solid line), atredshifts z=0,1,2,
respectively,fromtoptobottom.
theprojectionofthelenscenterinthesourceplane.Herewehave
setg=0intoEq.(60),i.e.weadoptasphericalpotential,andthe Theopticaldepthforabeamoflightfromasourceduetolens-
sourceisatzs=3,butverysimilarplotsareobtainedforzsinthe ingisproportional tothenumber densityof deflectorsmultiplied
range1 4.Ascanbeseen,themagnificationismaximizedwhen bythecrosssectionforagivenmagnification,integratedalongthe
÷
the source and the lens are aligned and Amax 26. If the lens is lineofsight.Twocompetingeffectsarethereforerelevant.Onone
≃
placedinsteadatzl=0.5,thenb=1.2arcsecandAmax 13. hand,thepathlengthtoasourceislargerforaL CDMmodel.On
≃
However, as already remarked, strongly lensed sources are the other hand, the structure formation histories within the stan-
usuallyexplained intermsofpotentialswithnon vanishing ellip- dardhierarchicalclusteringscenarioarealsodifferentforthedif-
ticity.Atypicalvalueforgcouldbe0.1.Inthiscasethesymme- ferent cosmologies. If we normalize the models to reproduce the
try of AE around r=0 is lost and the maximum magnification, localclusterabundance,thedensityoflowermassobjectsislower
Amax 12, significantly lower than in the spherical case, occurs intheL CDMmodel,becausethemassfunctionisflatter(e.g.Eke
≃
whenthelensandthesourceareoffsetby 0.7′′ alongthemajor etal.1996).ThisisshownintheupperpanelofFig.4,wherethe
∼
axisoftheellipse.TheresultsaredetailedinFig.2. massfunction[Eq.(1)]isplottedasafunctionofthehalomassat
We conclude that reasonable values of Amax for extended z=0.Thenumberoflow-massobjectskeepsincreasingwithred-
sourcesareintherange10–30.Thelowervalueisarelativelycon- shiftabove theL CDMmodel, whiletheopposite istrueforhigh
servativelowerlimit,easilyexceededforawiderangeofvaluesfor masses.
therelevantquantities,whilethelatterisobtainedonlyunderrather Thenextquestionisthen:whichisthemassrangecontributing
specialconditions. mosttotheopticaldepth?InFig.5weshow asafunctionofthe
lensmass,boththecrosssectionformagnificationsA>2andthe
lensmassfunctionforgivenvaluesofthesourceandlensredshifts.
A SIS profile is adopted. It may be noted that the magnification
8 NUMERICALRESULTSANDDISCUSSION
crosssectionissimilarforthetwomodels(althoughslightlyhigher
fortheflatCDMmodel).Theproduct ofthetwofunctionspeaks
8.1 Effectsfromcosmology
at masses between 1011 and 1012M for both a flat CDM and a
Toillustratetheeffectofthecosmological model onthemagnifi- L CDMmodel. ⊙
cationprobabilitydistribution p(A,z),weplotinFig.3theampli- Theeffectivemassofdarkmatterhaloscontributingmostto
tudea(z)ofthehigh-magnificationtailfortwodifferentcosmolo- stronglensingofasourcelocatedatzscanbeestimatedas:
Wgi0ems,=vi0z..3a.WflaethCavDeMadaonpdtedaSL ICSDpMrofimleosdfeolrwleintshesW a0nL d=as0m.7ooatnhd- M = dzl dMMdP(A,zs) , (64)
nessparametera s=0.9.Thetwomodelsarenormalizedtorepro- h i Z Z dzldM
ducethelocalabundanceofrichclusters:s 8=0.56W 0−m0.47 (Viana where P(A,zs) is the probability to have magnification >A. The
&Liddle1999). innerintegral,d M /dzl,isplottedasafunctionofzl,forzs=5,a
h i
(cid:13)c 0000RAS,MNRAS000,000–000
Gravitationallensing 9
1000
100
10
1
0.1
0.01
0.001
0.0001
0
0 1 2 3 4
Figure5.Numberdensityofdark-matterhalos(fromtheSheth&Tormen Figure6.Contributionsfromdifferentredshiftstotheeffectivelensmass
massfunction), andcrosssection formagnification A>2byaSISlens, (see text) fora fixed configuration of the lens system, SIS lenses, and a
inarbitraryunits,asafunctionofthelensmassinM .Dottedlinescor- L CDMmodel.
respondtoflatCDM,solidlinestoL CDM,bothnorm⊙alizedtothecluster
abundance.
SISlensprofileandaL CDMmodel,inFig.6.Asillustratedbythis
Figure,themaximumcontributiontothemagnificationprobability
comesfromthemassrange(1011–1012M )forwhichspaceden-
sitiesimpliedbyaCDMmodelareapprec⊙iablyhigherthaninthe 1
caseofaL CDMmodel,intherelevantredshiftinterval,ifthemod-
Solid lines: SIS
elshavetobeconsistentwiththeobservedclusterabundance.This 0.1 Dot-dashed lines: NFW
over-compensatesforthelargerpathlengthtoasourceinaL CDM
modelandexplainswhytheprobabilitydistributionofstrongmag-
0.01
nificationshasaloweramplitudeforsuchmodel.
It maybenoted (Fig.3)that thedifference betweenthetwo
0.001
cases decreases with increasing source redshift, since also the
L CDM model approaches an Einstein-de Sitter universe at high
0.0001
redshift.
Also shown in Fig. 3 is the amplitude a(z) for a COBE-
normalized standard-CDM model. The relatively large values for
thisquantityfor any source redshift areunrealisticsincetheyare
duetoamassfunctionofdarkhalosinconsistent withthecluster
abundance.
8.2 SISversusNFWprofiles
0 20 40 60 80
Figure7comparestheeffectofSISandNFWlensprofilesonthe
Amplification A
magnificationdistributionsforsourcesatzs=4andzs=7,includ-
ing the weak lensing effect for A<Acut. The magnification dis-
tributions are obtained by integrating over lens masses the cross
sections in the source plane, weighted by the mass distribution Figure7.MagnificationdistributionfromapopulationofSIS(solidlines)
[Eq.(1)].Theweak-lensingregime,responsibleformagnifications andNFWlenses(dot-dashed lines)forsourcesatredshifts zs=4(lower
belowAcut,givesrisetoaGaussianpeaknearA=1whosedisper- curves)andzs=7(uppercurves).
sionincreaseswithincreasingsourceredshift.Inthestrong-lensing
(cid:13)c 0000RAS,MNRAS000,000–000
10 PerrottaF.,BaccigalupiC., BartelmannM., DeZottiG., GranatoG.L.
0.1 1000
Strong lensing: 100
0.01 Solid lines: SIS
Dot-dashed lines: NFW
10
0.001
1
0.0001 0.1
0.01
0.001
0.0001
10
A
Figure8.High-magnification tail ofP(A).Magnifications areplotted on Figure9.Magnification crosssections s (A)(insquarearcsec)forA>2
alogarithmicscale.TheplotsrefertopopulationsofSIS(solidlines)and (uppercurves)andA>10,asafunctionofthehalomass.Thesourcesare
NFWlenses(dot-dashedlines)forsourcesatredshiftszs=4(lowercurves) atzs=5andthelensesatz=1.ThecrosssectionsareplottedforSIS(solid
oratzs=7(uppercurves). lines)andNFWhalos(dottedlines)
.
regime,theprobabilitydistributionisobtainedbysolvingEq.(12)
numericallyandinsertingtheresultinEq.(53).Theasymptoticbe- andrs 15h−1kpc,respectively,forh=0.65;theconcentrationis
haviour(cid:181) A−3 isreachedearlierbythemostdistantsources.The c 10≃.Theconvergencek forthetwoprofilesisshownintheup-
high-magnificationtailofFig.7isshownincloserdetailinFig.8. pe≃rpanelofFig.11asafunctionofhalocentricdistance.Unlikethe
Note that the plotted distribution of magnifications has a discon- surfacedensity,theconvergence dependsalsoonthegeometryof
tinuity in Acut, i.e. at the transition between the weak and strong thelenssystem.Here,thelensisatzl=1andthesourceatzs=5.
lensingregimes. Thetwoconvergenceprofilesarequitesimilar,buttheshearisalso
The two density profiles lead to slightly different magnifi- playing a fundamental role to determine the magnification distri-
cation distributions. In particular, the NFW lens ismore efficient bution.InthelowerpanelofFig.11,weplot,forthetwoprofiles,
thantheSISformoderatemagnifications(2<A<4),andlessef- det 1A, i.e. the image magnification µ of an image with impact
−
∼ ∼
ficient for high magnifications. Infact, NFW lenses havesmaller parameter r inthelensplane. Notethat,even ifsomeimagesare
high-magnification cross sections than SIS lenses of equal mass, demagnifiedbylensing,thetotalmagnificationofthesourceisal-
even if the average magnification is higher. This can be read off ways>1(e.g. Schneider et al.1992). Thecriticalcurvesbehave
Fig.9,whereweplotthecrosssectionformagnificationsA>2and differently.EventhoughtheNFWprofilehasasingularcore,ithas
A>10asafunctionofthehalomassforthetwomodels,keeping tangential andradialcriticalcurves(Bartelmann1996), whilethe
fixedtheconfigurationofthesystem.Forvirtuallyallhalomasses, SIShasonlyatangentialcriticalcurve(whosecausticdegenerates
s (10)NFW<s (10)SIS,whiles (2)NFW>s (2)SIS.Eventhoughthe toapointforallaxiallysymmetriclenses).Consequently,themax-
latterrelation failsfor very small lensmasses, it stillholds when imumimagenumberistwoforSISandthreeforNFWlenses.
thecrosssectionsareweightedwiththeappropriatemassfunction. Figure11showsthatthetangentialcriticalcurveofaSISoc-
As mentioned above, the bulk of the contribution tothe magnifi- cursatalargerradiusthanbothcriticalcurvesofanNFWhalowith
cationdistributioncomesfromalimitedmassrange.Theeffective equalmass.Thismeansthathighmagnificationsarefavoredinthe
massdefinedinSect.8.1isnearlyequalforSISandNFWprofiles, SISmodel,becausethecorrespondingcrosssectionsinthesource
namely 1011 12M .Although M depends (albeitweakly) on planeislarger.Ontheotherhand,theSISprofileyieldslowertotal
−
thelens∼redshift,mas⊙siveclustershneviercontributesubstantiallyto magnificationswhenthesourcelieswelloutsidetheoutercaustic
theintegrandinEq.(53)becausetheyareextremelyrare. inthesourceplane.TheNFWcrosssectionsforµtot 2generally
∼
TheSISandNFWdensityprofilesandthecorrespondingmass overcometheSISones,duetothefactthatthelensingpotentialis
encompassedwithintheradiusrfora1012M objectareshownin lesscurved for theNFWthan for theSISprofile, hence thisalso
Fig.10.Virialradiiareapproximatelyequal⊙inthetwocases.For occursformoremassivehalos.InFig.12,weplottheequilibrium
theNFWprofile,thevirialandscaleradiiarer 140h 1kpc configurationsfora1014M halo.TheNFWprofilehasvirialand
200 −
≃ ⊙
(cid:13)c 0000RAS,MNRAS000,000–000