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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

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

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

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