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DRAFTVERSIONFEBRUARY2,2008 PreprinttypesetusingLATEXstyleemulateapjv.6/22/04 EFFECTSOFELLIPTICITYANDSHEARONGRAVITATIONALLENSSTATISTICS DRAGANHUTERER,1,2,3 CHARLESR.KEETON3,4,5 &CHUNG-PEIMA6 DraftversionFebruary2,2008 ABSTRACT We studytheeffectsofellipticityin lensgalaxiesandexternaltidalshearfromneighboringobjectsonthe statisticsofstronggravitationallenses. ForisothermallensgalaxiesnormalizedsothattheEinsteinradiusis independentof ellipticity and shear, ellipticity reduces the lensing cross section slightly, and shear leaves it unchanged. Ellipticity and shear can significantly enhancethe magnificationbias, butonly if the luminosity functionofbackgroundsourcesissteep. Realistic distributionsofellipticityandshearlower thetotaloptical 5 depthbyafewpercentformostsourceluminosityfunctions,andincreasetheopticaldepthonlyforsteeplu- 0 0 minosityfunctions.Theboostintheopticaldepthisnoticeable(&5%)onlyforsurveyslimitedtothebrightest 2 quasars(L/L∗&10). Ellipticityandshearbroadenthedistributionoflensimageseparationsbutdonotaffect themean. Ellipticityandshearnaturallyincreasetheabundanceofquadruplelensesrelativetodoublelenses, n especiallyforsteepsourceluminosityfunctions,buttheeffectisnotenough(byitself)toexplaintheobserved a quadruple-to-doubleratio. With such small changes to the optical depth and image separation distribution, J ellipticity and shear have a small effect on cosmological constraints from lens statistics: neglecting the two 31 leadsto biasesof just∆ΩM =0.00±0.01and∆ΩΛ =- 0.02±0.01(wheretheerrorbarsrepresentstatistical uncertaintiesinourcalculations). 2 Subjectheadings:cosmology:theory—gravitationallensing v 0 4 1. INTRODUCTION To our knowledge, this conventional wisdom is based on 0 A circularly symmetric gravitationallens is a usefultheo- a few studies in which the analysis of ellipticity and shear 5 retical construct that, most likely, will never be observed in was subordinate to practical applications of lens statistics. 0 a cosmological setting.7 Every real cosmological lens will King&Browne (1996), Kochanek (1996b), Keetonetal. 4 (1997), and Rusin&Tegmark (2001) all computed the rela- have some small asymmetries either in its own mass dis- 0 tive abundancesof differentimage configurationsas a func- tribution (e.g., ellipticity), or in the distribution of objects / tionofellipticityand/orshear,forvariousassumptionsabout h near the line of sight (leading to a tidal shear). In fact, it p is well known that ellipticity and shear cannot be ignored the luminosity function of background sources. Along the - in models of individual observed strong lens systems (e.g., way, they necessarily computedthe effects of ellipticity and o shearonthelensingcrosssectionandmagnificationbias,but Keeton,Kochanek,&Seljak1997;Witt&Mao1997). r didnotexplicitlydiscussthem.Chae(2003)includedelliptic- t Nevertheless, most analyses of the statistics of grav- s ity in lens statistics constraints on cosmological parameters, a itational lenses have used symmetric lenses. The sta- buttheeffectswerebuiltintohisstatisticalmachineryandnot : tistical calculations offer enough intrinsic challenges that v presentedontheirown.Webelievethereispedagogicalvalue most authors have stuck to idealized spherical lenses, such i in isolating the statistical effects of ellipticity and shear and X as the singular isothermal sphere (SIS) or the general- studyingthemindetail. Itwouldbeusefultolayoutexactly izedNavarro-Frenk-White(GNFW;Navarro,Frenk,&White r howellipticityandshearaffectthelensingopticaldepth,and a 1997; Zhao 1996) profile (e.g., Narayan&White 1988; howthatmay(ormaynot)leadtobiasesincosmologicalcon- Fukugita&Turner1991;Kochanek1995,1996a;Maozetal. straints. Furthermore,atleasttwo otherissues deservetobe 1997; Keeton&Madau 2001; Sarbu,Rusin,&Ma 2001; studiedaswell: theeffectsofellipticityandshearonthedis- Takahashi&Chiba 2001; Li&Ostriker 2002; Davisetal. tributionoflensimageseparations;andthedependenceofthe 2003;Huterer&Ma2004;Kuhlenetal.2004;Mitchelletal. various statistical effects on the luminosity function (LF) of 2004). Conventionalwisdomholdsthatthestatisticaleffects thebackgroundsources. Wewillshowthat,whilenotwrong, ofellipticityandshearareconfinedtotherelativenumbersof the conventional wisdom is somewhat limited and there are double and quadruple lenses, and that symmetric lenses are effectsofellipticityandshearonlensstatisticsthataresubtle adequateforapplicationssuchasderivingcosmologicalcon- butinteresting. straints. We focus on lensing by galaxies, by which we mean sys- 1 Department of Physics, Case Western Reserve University, Cleveland, temswithasingledominantmasscomponentthatcanbeap- OH44106 proximated as an isothermal ellipsoid. The isothermal pro- 2 Kavli Institute for Cosmological Physics, University of Chicago, file describes early-type galaxies remarkably well on the 5– Chicago,IL60637 10kpcscalesrelevantforstronglensing(e.g.,Rixetal.1997; 3 Astronomy & Astrophysics Department, University of Chicago, Gerhardetal. 2001; Rusin&Ma 2001; Treu&Koopmans Chicago,IL60637 4HubbleFellow 2002;Koopmansetal. 2003; Rusinetal. 2003). Lens statis- 5 Physics&AstronomyDepartment,RutgersUniversity, Piscataway,NJ tics are rather different for groups and clusters of galaxies 08854 modeled with GNFW profiles, and that parallel case has re- 6DepartmentofAstronomy,UniversityofCalifornia,Berkeley,CA94720 centlybeenstudiedbyOguri&Keeton(2004). 7 Theonlyexception isGalactic microlensing, wheretheseparationbe- tween stars is large enoughcompared withtheir Einstein radii that anon- binarylensiswelldescribedasasymmetricpoint-masssystem. 2. METHODOLOGY 2 2.1. Generaltheory where the second factor depends only on the ellipticity and sheardistributions,whilethefirstfactordependsonlyonthe Theprobabilityfora sourceatredshiftz tobe lensedcan s sourceredshiftandthedeflectorpopulation. Ifweonlywant bewrittenas the change in the optical depth produced by ellipticity and τ(z )= 1 dV dσ dn de p (e) d2γ p (γ,φ ) shear,thenwecansimplywrite s e γ γ 4π dσ Z×ZmZultd~u ΦΦsrcsrZ(cL(L/µ)) . Z (1) ττ0 =Z de pe(e)Z d2γ pγ(γ,φγ) BAˆ(eB,Aˆγ0,φγ) , (4) ˆ Thefirstintegralisoverthevolumeoftheuniverseouttothe whereBA andτ arethebiasedcrosssectionandtheoptical 0 0 source.Thesecondintegralisoverthepopulationofgalaxies depthforthesphericalcase. that can act as deflectors. For isothermal lenses the impor- Working in dimensionless units also simplifies the study tantphysicalparameteristhevelocitydispersion,sothemost of image separations. Even if all galaxies are SIS, the dis- usefuldescriptionofthegalaxypopulationisthevelocitydis- tribution of image separations will be fairly broad because persiondistributionfunction(dn/dσ)dσ,orthenumberden- there is a range of lens galaxy masses and redshifts (see, sityofgalaxieswithvelocitydispersionbetweenσandσ+dσ e.g., Kochanek 1993a). However, the dimensionless separa- (seeMitchelletal.2004). Thethirdintegralisoveranappro- tion ∆θˆ=∆θ/θ is always ∆θˆ=2 for an SIS lens, so the E priatedistribution pe fortheinternalshape(ellipticity)ofthe dimensionlessimageseparationdistributionp(∆θˆ)isjustaδ- lensgalaxy. (Withoutlossof generality,we can workin co- functionwhenthe ellipticityand sheararezero. Thismeans ordinatesalignedwiththemajoraxisofthegalaxysowedo that studying p(∆θˆ) is the simplest way to identify changes not need to consider the galaxy position angle.) The fourth totheimageseparationdistributioncausedbyellipticityand integralisoveranappropriatedistribution p fortheexternal γ shear. For fixed ellipticity and shear, the distribution can be tidal shear caused by objects near the lens galaxy; this inte- formallywrittenas gralistwo-dimensionalbecauseshearhasbothanamplitude a(γn)gualnadrapodsiirteicotnio~un(oφfγt)h.eFsionuarllcye,tihnethfiefthsoiunrtceegrpallainseo,vaenrdthies p(∆θˆ|e,γ,φγ)= d~uΦΦsrc(L(L/µ))δ ∆θˆ- ∆θˆ(~u;e,γ,φγ) , limited to the multiply-imagedregion. In the last integrand, Zmult src h (i5) µisthelensingmagnification,Φsrc(L)isthecumulativenum- where ∆θˆ(~u;e,γ,φ ) is the dimensionless image separation ber density of sources brighter than luminosity L in the sur- γ vey,andthefactorΦ (L/µ)/Φ (L)accountsforthe“mag- producedfora sourceatposition~ubya lenswith the speci- src src fiedellipticity andshear. The fullimageseparationdistribu- nification bias” that producesan excess of faint sourcesin a tioncanthenbefoundbyintegratingoverappropriatedistri- flux-limitedsurveyduetolensingmagnification(Turneretal. butionsofellipticityandshear. Notethatwedonotactually 1984). (The role of the limiting flux or limiting luminosity needtointegrateoverthemassesandredshiftsofthedeflector willbediscussedin§2.3below.)Thedifferentialprobability forhavingalenswithimageseparation∆θ canbecomputed populationin order to compute changes to the optical depth andtheimageseparationdistribution. byinsertingaDiracδ-functionineq.(1)toselecttheparame- tercombinationsthatgiveseparation∆θ. Inotherwords,we canthinkofthe(normalized)imageseparationdistributionas 2.2. Theisothermalellipsoidwithshear p(∆θ)=τ- 1∂τ/∂∆θ. We first discuss isothermal ellipsoids without an external ThelensingcrosssectionAandthemagnificationbiasfac- shear, and then discuss propertiesof shear at the end of this tor B are often computed separately (see Chae 2003 for the subsection. mostrecentexample).Accordingtoeq.(1),however,thetwo Theprojectedsurfacemassdensityforanisothermalellip- quantitiesarecloselylinked.Weprefertokeepthemtogether soid,writteninpolarcoordinates(r,φ)andexpressedinunits andcomputetheproduct ofthecriticaldensityforlensing,is BA≡ d~u Φsrc(L/µ) , (2) Σ b 1+q2 1/2 Φ (L) κ(r,φ)= = , (6) Zmult src Σcrit 2r(cid:20)(1+q2)- (1- q2)cos2φ(cid:21) which we call the “biased cross section.” The biased cross where q ≤ 1 is the axis ratio, and the ellipticity is section depends on both the lens model parameters and the e = 1- q. (Recall that we are working in coordinates sourceLF. aligned with the major axis of the galaxy.) The ra- Aconvenientfeatureofisothermallensesisthatthephysi- dius r and the parameter b both have the dimensions of calscaledecouplesfromthelensingproperties.Allofthede- length, and may be expressed as physical lengths (e.g., pendenceonz ,z,andσiscontainedinthe(angular)Einstein s l kiloparsecs) or angles on the sky (radians or arcseconds); radius θ , so when we work in units of θ nothing depends E E we work in angular units. The lensing properties of explicitlyontheseparameters. Asaresult,thedimensionless ˆ an isothermal ellipsoid are given by Kassiola&Kovner biasedcrosssectionBA≡BA/θ2 dependsonlyontheelliptic- E (1993), Kormann,Schneider,&Bartelmann (1994), and ityandshear(andimplicitlyonthesourceLF).We canthen Keeton&Kochanek(1998). rewriteeq.(1)as TheEinsteinradiussetsthelensingscale,soitisusefulto 1 dn determine its value. Consider the deflection α0(r) produced τ(zs)= 4π dV dσ dσ θE2(zs,zl,σ) × bythemonopolemomentofthelensgalaxy, (cid:26) Z Z (cid:27) de p (e) d2γ p (γ,φ )BAˆ(e,γ) , (3) α (r)= 1 rdr′ 2πdφr′κ(r′,φ)= Mcyl(r) , (7) e γ γ 0 πr πrΣ (cid:26)Z Z (cid:27) Z0 Z0 crit 3 whereM (r)istheprojectedmassina cylinderofradiusr. cyl The Einstein radius is defined by α (θ )=θ . This defini- 1.2 0 E E tion reduces to the standard Einstein radius in the spherical all oblate case, and it is the quantitythat seems to be most relevantin models of nonspherical lenses (e.g., Cohnetal. 2001). For 1.1 theisothermalellipsoid,wefind ) 0 half-half θbE = π1 2(1+q- 2) 1/2 K 1+q- 2 , (8) e) / b( 1 fixed Q whereK(x)istheelli(cid:2)pticintegr(cid:3)alofth(cid:0)efirstk(cid:1)ind. Forrefer- b( E encewenotethatthisfunctioncanbeapproximatedby 0.9 all prolate θ (e) E =exp (0.89e)3 , (9) b 0.8 whichisaccurateto<1%fore≤(cid:2) 0.53an(cid:3)dto<4%fore≤0.9. 0 0.1 0.2 0.3 0.4 0.5 Inpractice,however,weusetheexactresult. SIE ellipticity We must specify how to normalize the model, or how to choose the parameter b. For a spherical galaxy, b simply FIG. 1.— Changeintheisothermalellipsoidbparameterasafunctionof ellipticity.Thetwodashedcurvesshowthedynamicalnormalizationifallha- equals the Einstein radius and is related to the velocity dis- losareassumedtobeoblateorprolate.Thedot-dashedcurveshowsthecase persionby whenhalfofthehalosareassumedtobeoblateandhalfprolate. Thesolid σ 2 D curveshowstheresultwhentheEinsteinradiusisfixedtobeindependentof b=θE =4π ls , (10) ellipticity,whichiswhatweassumeinthispaper. c D os (cid:16) (cid:17) where D and D are angular diameter distances from the os ls observertothesourceandfromthelenstothesource. Fora relativetothelensgalaxy;theshearamplitudewouldbeγ= nonspherical galaxy the situation is less straightforward. If b /(2r ) andthe sheardirectionwouldbeφ =φ . External we seek a dynamical normalization in terms of a measur- 0 0 γ 0 shear does not contribute to the local surface mass density, able stellar velocity dispersion, then we must worry about so it does not affectthe monopoledeflection or the Einstein complicationsinvolvingthehaloshapeandprojectioneffects radius. (Keetonetal. 1997; Keeton&Kochanek 1998; Chae 2003). ConsiderthedynamicalnormalizationshowninFigure1(fol- 2.3. Sourceluminosityfunctions lowing Chae 2003). At a typical ellipticity e≈0.3, b could rise by 7%(comparedto the sphericalvalue) if all halos are The number density of sources with luminosity be- oblate,orfallby7%ifallhalosareprolate.Somedissipation- tween L and L+dL is given by the luminosity function less numericalsimulations have predicted roughly compara- [dφ (L)/dL]dL. The quantity of interest for lens statistics src blenumbersofoblateandprolatehalos(Dubinski&Carlberg (see eq. (1)) is the cumulative number density of sources 1991; Jing&Suto 2002), which would yield a b value less brighterthanL,or than 1% higherthan the sphericalvalue (for e=0.3). How- ∞dφ (L′) ever,theshapedistributionislikelytobeaffectedbyhydrody- Φ (L)= src dL′. (12) namics(e.g.,Kazantzidisetal.2004),soitisnotunderstood src ZL dL′ indetail(insimulations,letaloneinreality). Inotherwords, WeconsidermodelLFsappropriatetobothradioandoptical the dynamical normalization appears to be small but uncer- surveys. tain,andimpossibletocomputeprecisely. ThesimplestmodelLFisafeaturelesspowerlaw,φ (L)∝ src An alternate approach is to fix the Einstein radius to be L- β. Inthiscasethebiasedcrosssectionsimplifiesto independent of ellipticity (and shear; see below). This seems reasonable, because the Einstein radii of observed BA= µβ- 1d~u≡ µβ- 1 p(µ)dµ, (13) lenses can generally be determined in a model-independent way to a few percent accuracy (e.g., Cohnetal. 2001; Zmult Z Muñoz,Kochanek,&Keeton2001),andbecauseitkeepsthe where p(µ) is the distribution of magnifications for lensed sources. Several points are worth mentioning. First, with massproperties(theaperturemass)independentofellipticity andshear. ThisnormalizationisalsoshowninFigure1,andit β→1+themagnificationweightingfactorbecomesunityand we recover the simple lensing cross section with no magni- istheoneweadopt. However,itisimportanttokeepinmind fication bias.8 Second, because the magnification distribu- thatthereisanirreducibleuncertaintyofafewpercentinour analysisassociatedwiththenormalization. tion generically has a power law tail p(µ)∝µ- 3 for µ≫1 (see Schneider,Ehlers,&Falco 1992), the integral diverges Objectsinthevicinityofthelensgalaxycreatetidalforces thataffectthe lenspotential. Thecontributionis oftenmod- for β ≥3 and the biased cross section is well defined only eledasanexternalshearwhosecontributiontothelenspoten- for β < 3. Finally, a power law is featureless so the biased cross section does not depend on the particular flux or tialis Φ (r,φ)=- γr2cos2(φ- φ ), (11) luminosity limit of a survey. A power law LF is a good shear 2 γ model for radio surveys. For example, the largest existing lenssurveyistheJVAS/CLASSsurveyofflat-spectrumradio where(r,φ)arepolarcoordinates,γisthedimensionlessshear amplitude,andφ isthedirectionoftheshear.Asanexample, γ 8Wewriteβ→1+,meaningthatβapproachesunityfromabove,because consider the shear produced by an isothermal sphere galaxy forβ≤1thecumulativeLFintegral(eq.(12))formallydiverges. Neverthe- with Einstein radius b0 that lies at polar coordinates(r0,φ0) less,thebiasedcrosssectionintegral(eq.(13))remainswelldefined. 4 sources(Myersetal.2003;Browneetal.2003),andithasan LF that is well described by a power law with β ≈2.1 (see Rusin&Tegmark2001;Chae2003). 101 g =0 Futurelenssamplesarelikelytobedominatedbyoptically- g =0.2, doubles selectedquasarlensesfoundindeepwide-fieldimagingsur- g =0.2, quads ) vtieoynso(ef.gth.,eKquuhalseanreLtaFl.is20a04l)o.ngW-sthainlediancgcuprraotbeledmet,errmecinena-t mg10 100 o evidence favors the double power law form proposed by (l Boyle,Shanks,&Peterson(1988), d dφ(L,z) φ∗ dL dA / 10-1 dL= , (14) dL [L/L∗(z)]βl+[L/L∗(z)]βh L∗(z) where the break luminosity L∗ evolves with redshift as (Madauetal.1999) 10-20 0.5 1 1.5 2 log (m ) L∗(z)=L∗(0)(1+z)αs- 1 eζezξ(z1++eeξξzz∗∗) , (15) 10 FIG.2.— Magnificationdistributionsforsphericaldeflectorswithoutshear wherethequasarspectralenergydistributionisassumedtobe andwithshearγ=0.2.Thecurvesarenormalizedsothattheareaundereach a power law, fν ∝ν- αs. With this LF, the biased cross sec- curveisthecorrespondingcrosssectioninunitsofθE2.Fortheshearcasewe tionclearlydependsonthebrightandfaintslopesβ andβ, showthedistributionsfordoublesandquadruplesseparately. Notethatthe oanndsoalusroceonretdhsehilfimt tiotinthgeluemxtiennotstihtyatLtchute(sze)/qLu∗a(nz)t.itiIetshddeeppeenndlds mquiandimruupmlesmitaigsnµifi4c,maitnio=n2f/o[rγd(o1u- bγle2s)]is(Fµin2,cmhinet=a2l./2[(010+2)3;γso)(1µ-2,mγi)n],=w1h.5il6eafnodr µ4,min=10.4forthecaseγ=0.2shownhere.Thedistributionsasymptoteto onredshift. We adoptthe modelfromFanetal. (2001)with A(µ)∝µ- 3athighmagnifications(Schneideretal.1992). bright-end slope β =3.43 at z<3 and β =2.58 at z>3, h h andfaint-endslopeβ =1.64atallredshifts(Wyithe&Loeb l 2002). nearby clusters by Jørgensenetal. (1995). The distribution Ifwewantedtocomputestatisticsforrealquasarlenssur- has mean hei=0.31 and dispersion σ =0.18, and there are veys, we would need to adopt an appropriate limiting mag- e nogalaxieswithe&0.8. Althoughthemeasuredellipticities nitudeandcomputethelimitingluminosityLcut(z)/L∗(z)asa describetheluminositywhilewhatweneedforlensingisthe functionofredshift. Thiswouldrequirespecifyingthepass- ellipticity of the mass distribution, this is probably the best band, computingK-corrections,and otherdetails thatwould we can do at the moment. In any case, it seems reasonable muddythewaters.Sincethegoalisconceptualunderstanding to think that the ellipticity distributionsfor the light and the oftheeffects, webelievethatitissimplerandmoreinstruc- mass may be similar (see Rusin&Tegmark 2001). For the tivetoworkwithaluminositycutLcut/L∗. Inthiscase,wedo shear, Holder&Schechter (2003) estimate that the distribu- notneedtospecifytheparametersφ∗,L∗(0),z∗,ζ,andξ. tion of shear amplitudes derived from simulations of galaxy formation can be described as a lognormaldistribution with 2.4. Numericaltechniques medianγ=0.05anddispersionσ =0.2dex;thisdistribution γ Wecomputetheintegralsineqs.(2)–(5)usingMonteCarlo is also broadly consistent with the shears required to fit ob- techniques.First,forfixedellipticityandshearweplace105– servedlenses. Asaruleofthumb,ashearγ∼0.1iscommon 106randomsourcesinthesourceplane,inthesmallestcircle forlensesinpoorgroupsofgalaxies,andtheshearcanreach enclosingthe caustics. We solve the lens equationusing the γ ∼0.3 for lenses in rich clusters (e.g., Keetonetal. 1997; gravlenssoftware(Keeton2001)todeterminethenumberof Kundicétal.1997a,b;Fischeretal.1998;Kneibetal.2000). imagesandtheirpositionsandmagnifications. Wedefinethe Weassumerandomshearorientations. image separation to be the maximal separation between any twoimagesinthesystem,∆θ≡max|θ~ - θ~|;thisisaconve- 3. THEOPTICALDEPTH i j nient, observable, and well-defined quantity that is indepen- Before determining the effects of ellipticity and shear on dentofthenumberofimages. thelensingopticaldepth,itisinstructivetoconsiderfirsthow Weseparatethelensesintothreestandardclassesbasedon theyaffectthesourceplane. Thereisonlyasmallchangein theimagemultiplicity:“doubles”havetwobrightimages,one thelensingcrosssection.Infact,forisothermalgalaxiesshear withpositiveparityandonenegative,plusafaintcentralim- hasnoeffectontheradialcausticandhenceonthecrosssec- agethatisrarelyobserved;“quads”havefourbrightimages, tion.9 Ellipticity (or any other internal angular structure) in twopositiveandtwonegativeparity,plusafaintcentralimage isothermalgalaxieschangesthe causticsin sucha way as to that is rarely observed; and “naked cusps” have three bright reducethe cross section, as explainedin the Appendix. The images, either two positive parity and one negative or vice maineffectofincreasingellipticityorshearistolengthenthe versa. Weusetheclassificationsdirectlyonlywhenstudying tangential caustic, which enlarges the phase space for large thequadruple-to-doubleratio(§5). Theclassificationoffers magnifications and raises the tail of the magnification dis- afringebenefit: wecanidentifynumericalerrorsassystems tribution, as illustrated in Figure 2. In particular, we see a that do not fit into any of the classes (because, for example, sharp increase in the cross section for producing magnifica- the software failed to find one of the images). We estimate thatthenumericalfailurerateis<10- 4. 9Shearcanaffectthecrosssectiononlyintherarecasethatthetangential Next,whereappropriateweintegrateoverdistributionsfor causticpiercestheradialcaustictoformnakedcusps(e.g.,Schneideretal. 1992). ForSIS+shear models, this happens only when the shear is large, ellipticityandshear. Fortheellipticity,weadoptthedistribu- γ>1/3. Inthiscase,thereissome(small)multiply-imagedregionoutside tionofellipticitiesmeasuredfor379early-typegalaxiesin11 theradialcaustic. 5 tionslargerthantheminimummagnificationforaquadruple tionsweobtainthenetimageseparationdistributionshownin lens(seecaption). Figure8. Theaveraginghassmoothedoutthesharpfeatures We now examine the dependenceof the biased cross sec- seen in Figure 6 when the ellipticity and shear were fixed. tionBA onellipticity(Figure3) andshear(Figure4). When ThenetdistributionisnearlyGaussian,withmean∆θˆ=2.01 the LF is a power law with β →1+ there is no magnifica- and scatter σ∆θˆ=0.18 for a power law LF with β =2.1, or tciroonssbsiaesc,tiaonnd. EFvigeunrwei3thilmluasgtrnaitfiecsahtioownbelilaisp,teiclliitpytirceidtiuecseuspthtoe ∆θˆ≈2.01andσ∆θˆ≈0.19forvariouscasesofthequasarLF. Inotherwords,ellipticityandshearbasicallyleavethemean e∼0.5 do not affect the biased cross section by more than image separation unchanged but create an additional scatter 10%unlessthesourceLFisverysteep(e.g.,theverybright- of10%,andtheseresultsareinsensitivetothesourceLF. estquasars, Lcut/L∗ &100whenβh=3.43). Figure4 shows thatshear causesa strongerincrease in the biased crosssec- 5. QUADRUPLE-TO-DOUBLERATIO tion, butwe mustrememberthatrealistic shearsare γ.0.1 Wenextconsiderhowellipticityandshearaffectthenum- andonlylensesinclustersexperiencelargeshearsofγ∼0.2– ber of lenses with different image configurations. While an 0.3.Thus,thetypicalchangeinthebiasedcrosssectiondueto SISlensalwaysproducestwoimages,increasingellipticityor shearisagainnomorethan10%unlesstheLFisverysteep. shear leads to increasing probability for configurations with To compute changesin the full optical depth we integrate four images. Furthermore, large ellipticities (e>0.606) or the biased optical depth over appropriatedistributions of el- shears (γ > 1/3) can lead to “naked cusp” configurations lipticityandshear(seeeq.(4)). TheresultsareshowninFig- withthreebrightimages(e.g.,Keetonetal.1997). Nearlyall ure 5. Without magnification bias (β →1+), ellipticity and known lenses with point-like images are doublesor quadru- shear reduce the optical depth very slightly (by ∼0.6%, al- ples; among ∼80 known lenses there is only one candidate though the statistical uncertainty from our Monte Carlo cal- nakedcusplens(APM08279+5255;Lewisetal.2002). culations is ∼0.3%) With magnification bias and a shallow Figure 9 shows that the quadruple to double ratio rises source LF, ellipticity and shear can reduce the optical depth monotonicallywith ellipticity or shear. For the CLASS LF, byupto∼2.5%relativetothesphericalcase. Forpowerlaw the ratio is ∼20% for typical ellipticities e ∼0.3 or shears LFs, only when β ≥ 2.2 is there an increase in the optical γ∼0.1. Ourresultsagreewellwithpreviousanalyses(e.g., depth, and we must have β ≥2.5 (β ≥2.6) in order for the Rusin&Tegmark2001;Finchetal.2002). increase to be more than 5% (10%). For the quasar LF, the We can now estimate the expected number of quadruples increaseexceeds5%onlyifthebrightendissteep(βh=3.43) (andcusp triples) by averagingoverour fiducialdistribution andthesurveyislimitedtobrightquasars(Lcut/L∗&10). of ellipticity and shear. The results are shown in Figure 10 Inpractice,theseresultsmeanthatellipticityandshearare forboththepowerlawLFandthequasarLF.FortheCLASS important for the optical depth only in surveys that are re- LF(β=2.1),thenetquadruple-to-doubleratioisabout0.35, strictedtothebrightestquasars. Theyarenotverysignificant whilethetriple-to-doubleratioisonlyabout0.01. Thenum- forthesortsofdeepopticalsurveysnowunderwaythatprobe berofquadruplevs.doublesystemsintheCLASSstatistical wellbeyondthebreakinthequasarLF. sample is 5 vs. 7; the ratio is twice as large as our predic- 4. THEIMAGESEPARATIONDISTRIBUTION tion. We therefore agree with Rusin&Tegmark (2001) in concluding that ellipticity and shear alone cannot easily ex- We now turn to the distribution of lens image separations plain the high number of quadruples. Additional effects are and how it is affected by ellipticity and shear. First, we re- required, which are probablyrelated to lens galaxy environ- callseveralbasicfacts. Eveninsphericalmodelsthedistribu- ments.Shearisonlyalow-orderapproximationtothelensing tionofdimensionedimageseparationswillhavesomenatural effects of objects near the lens galaxy. Recent studies have spread because of the range of galaxy masses and redshifts; but the distribution of dimensionless separations ∆θîs a δ- shownthatincludinghigher-ordereffectsfromsatellitegalax- ies(Cohn&Kochanek2003)orextendedgroupsofgalaxies function at ∆θˆ=2. To highlight changes in the separation (Keeton&Zabludoff2004) aroundthelenscansignificantly distribution,itisthereforeusefultofocusonthedistribution boostthequadruple-to-doubleratio. of∆θˆ. Also,asdiscussedin§2.4,wedefinetheseparationto Figure 10 shows that surveys targeting lensed quasars are bethemaximaldistancebetweenanypairofimages. expected to have a low quadruple-to-doubleratio unless the Figure6showsthedistributionof∆θˆforseveralvaluesof brightendoftheLFissteep(βh=3.43)andthesurveyislim- ellipticity (upperpanel) and shear (lower panel). The distri- ited to brightquasars(Lcut/L∗ &10). This predictioncould, bution has an interesting shape that peaks at the ends and is ofcourse,beanunderestimatebecausewehaveneglectedthe lowinthemiddle. Ithasasharpcutoffatthehighend,while higherordereffectsfromlensenvironments. atthelowendithasasharpdropfollowedbyasmalltailto 6. EFFECTSONCOSMOLOGICALCONSTRAINTS lowervalues.Thepeakscorrespondtosourcesneartheminor While the changes in the optical depth and image separa- andmajoraxesofthelenspotential. Astheellipticityorshearincreases,thedistributionof∆θˆ tiondistributioncausedbyellipticityandshearseemmild,it isimportanttoquantifyhowtheyaffectoneofthemainappli- broadens and its mean shifts. To quantify these effects, we compute the mean separation h∆θî and the spread σ∆θˆ = ceatetiros.nsOonfeleanpsprsotaatcihstiwcso:ulcdonbsetrtaoinmtsoodnifycothsmeoanloaglyicsaelspoafrarema-l (h∆θˆ2i- h∆θî2)1/2, and plot them as a function of elliptic- lenssamplestoincludethefulleffectsofellipticityandshear ityorshearinFigure7. Theincreaseinthemeanandscatter (buildingupontheanalysisofChae2003). Suchanapproach, are small for all ellipticities, and are both <20% for all but however,wouldbelimitedbyPoissonuncertaintiesincurrent the strongest shears (γ &0.3) felt by lenses in cluster envi- lenssamples(e.g.,CLASShasjust13lenses),bysystematic ronments.Nevertheless,itisinterestingthatshearproducesa uncertaintieswhere modelsmayor may notbe correct(e.g., netbiastowardlargerimageseparations. evolution in the lens galaxy population; see Mitchelletal. Finally,byaveragingovertheellipticityandsheardistribu- 2004),andbysystematiceffectsthatareknowntobepresent 6 1.4 1.4 b =2.6 LLcut//LL* == 110 thick lines: b h=3.43 1.3 b =2.3 1.3 Lcut/L* = 100 thin lines: b =2.58 b =2.0 cut * h b =1.0 0 0 A) 1.2 A) 1.2 B B ( ( A / 1.1 A / 1.1 B B 1 1 0.9 0.9 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 ellipticity ellipticity FIG. 3.— Enhancementinthebiasedcrosssectioncausedbyellipticity. Theshearissettozero. (a,left)ResultsfordifferentpowerlawsourceLFs. Recall thatwithβ→1+thereisnomagnificationbias,andthattheCLASSradiosurveyhasβ≈2.1.(b,right)ResultsforthemodelquasarLF,fordifferentvaluesof thelimitingluminosity(Lcut/L∗)andthebright-endslope(βh);thefaint-endslopeisfixedatβl=1.64. Thejaggednessintheuppercurvesisduetostatistical noise,becauseforsteepLFsthemagnificationbiasisstrongandtheresultsaredominatedbyrareextreme-magnification systems. Weestimatethestatistical errorsintheuppercurvestobe∼3%,andmuchsmallerfortheothercurves. 1.4 1.4 L /L = 1 cut * b =2.6 L /L = 10 b =2.3 Lcut/L* = 100 1.3 b =2.0 1.3 cut * b =1.0 0 0 A) 1.2 A) 1.2 B B ( ( A / 1.1 A / 1.1 B B 1 1 thick lines: b =3.43 thin lines: b =2.58 h h 0.9 0.9 0 0.1 0.2 0.3 0 0.1 0.2 0.3 shear shear FIG. 4.— SimilartoFigure3,butforshear. Theellipticityissettozerohere. Thestatisticalerrorsintheuppercurvesare∼10%,andmuchsmallerforthe othercurves. 1.3 1.3 b =3.43 b =2.58 1.2 1.2 0 0 t/ 1.1 t/ 1.1 t t 1 1 0.9 0.9 1 1.5 2 2.5 0.1 1 10 100 b L / L cut * FIG. 5.— EnhancementintheopticaldepthasafunctionoftheluminosityfunctionslopeβforpowerlawLFs(left)orthelimitingluminosityLcut/L∗for quasarLFs(right).Intheleftpanel,thesolidlineshowsthefiducialresultwhilethedottedlinesindicatethestatisticaluncertaintyfromourcalculation. 7 3 20 SIS SIE, e=0.2 2.5 15 SIE, e=0.4 10 2 5 1.5 0 15 SIS SIS+XS, g=0.1 1 10 SIS+XS, g=0.2 0.5 5 0 0 1 1.5 2 2.5 3 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Dq / q Dq / q E E FIG.6.— Histogramsofthedimensionlessimageseparation∆θˆ=∆θ/θE, FIG. 8.— Netimage separation distribution after averaging overellip- fortwo values ofellipticity (top panel) and shear (bottom). Foraspheri- ticityandshear. Thehistogramisnormalizedtounitarea. Theresultsare clianle.leTnshethheisdtiosgtrriabmutsiocnonistaainδa-lfluinmctaigoenmatu∆ltiθpˆl=ici2ti,eisnd(ii.cea.,tebdotbhydtohuebvleesrtiacnadl adgisatirnibsuhtioownnisfonreathrleyCGLaAusSsSiaLn,Fwbiuthtamreeanno∆tvθêr=y2s.e0n1siatinvdestcoatthteirsσch∆oθîc=e.0T.1h8e. quadruples). Theresults areshownfortheCLASSLF(apowerlaw with β=2.1),buttheyarenotverysensitivetothischoice. 1 3.5 Ellipticity 0.8 Shear 3 EShlleiparticity atio ^ 2.5 <Dq ^> ble r 0.6 Dq u s> or 1.25 ad / do 0.4 ^ u q 0.2 Dq< 1 s Dq ^ 0.5 0 0 0.1 0.2 0.3 0.4 0.5 e or g 0 0 0.1 0.2 0.3 0.4 0.5 e or g FIG. 9.— Quadruple-to-double ratioasafunctionofellipticityorshear, assumingtheCLASSLF(apowerlawwithβ=2.1). FIG.7.— Meanandspreadoftheimageseparationdistributionasafunc- tionofellipticity orshear. TheimageseparationsareinunitsofθE. Asin Figure6,theresultsareshownfortheCLASSLFbutarenotverysensitive tothischoice. opticaldepthτ(z )toobtaintheredshiftdistributionof s thelensedsources(seeMitchelletal.2004). inthedatabuthavenotyetbeenstudied(e.g.,havingmultiple • Thelensredshiftisdrawnfromthedistribution p(z)∝ l lensgalaxies). Webelievethatitismoreinstructivetocreate (D /D )2 dV/dz, where the factor of (D /D )2 ls os l ls os mocklenssurveysthatmimicCLASSbutallowustoisolate comes from the factor of θ2 in the lensing cross sec- E theeffectsofellipticityandshear. Specifically,wecreatesur- tion(seeeq.(10)). veysthatincludeellipticityandshear,andthenanalyzethem using standard spherical models in order to uncover biases • The velocity dispersion is drawn from p(σ) ∝ that result from neglecting ellipticity and shear. We create σ4 dn/dσ, where the factor of σ4 comes from θ2 in E mocksurveyswith1000lensesinordertominimizePoisson the lensing cross section. We use the velocity disper- uncertainties. We use a Monte Carlo approach, drawingpa- siondistributionfunctiondn/dσderivedbyShethetal. rameter valuesfrom appropriateprobabilitydistributions(as (2003)forearly-typegalaxiesintheSloanDigitalSky indicatedineq.1). Specifically: Survey.Forsimplicity,weassumethatthevelocitydis- persion distributiondoesnotevolve with redshift. We • AsubsetofsourcesintheCLASSsurveyhasaredshift could add evolution to both the creation and analysis distributionthatcanbetreatedasaGaussianwithmean ofthemocksurvey(see,e.g.,Mitchelletal.2004),but hzsi=1.27 and width σz =0.95 (Marlowetal. 2000), thatwouldjustcomplicatematters. and this is usually taken as a model for the redshift distribution of the full survey (e.g., Chae 2003). The • WeusetheellipticitydistributionfromJørgensenetal. Gaussianismodifiedbytheredshiftdependenceofthe (1995), the shear amplitude distribution from 8 1.5 4 quad / double b =3.43 triple / double o b =2.58 i 3 t a 1 r e l o b ati ou 2 r d / 0.5 d a u 1 q 0 0 1 1.5 2 2.5 0.1 1 10 100 b L / L cut * FIG.10.— Quadruple-to-doubleratio,asafunctionoftheslopeβofpowerlawLFs(left)orthelimitingluminosityLcut/L∗forquasarLFs(right).Theresults areobtainedbyaveragingoverthedistributionsofellipticityandsheardiscussedinthetext. Intheleftpanel,wealsoshowtheratiooftriple(ornakedcusp) lensesasthedashedcurve. Holder&Schechter (2003), and random shear di- rections. • Whendrawingrandomsourcepositions,weusemagni- ficationbiasappropriatetotheCLASSsurvey(apower law with β =2.1) since it is the most commonlyused surveyincurrentlensstatisticanalyses. Given the parameters we can compute the observables for eachmocklens: the sourceandlensredshiftsandtheimage separation. The other key observable is the total number of sourcesin thesurvey. We usetheopticaldepthtodetermine the number of sources needed to obtain 1000 lenses, which istypically∼8×105. Wedistributethesesourcesinredshift usingtheGaussiangivenabove. Wethenanalyzethemocksurveywithstandardmaximum likelihood techniques. Assuming complete data — knowl- edgeoftheimageseparationandthelensandsourceredshifts for lens systems, and the redshift distribution of non-lensed sources—weusethelikelihoodfunction (Npred)Nobs e- Npred Nlens 1 ∂2τ L= × . (16) N ! τ(z ) ∂z ∂∆θ obs s,i l,i i Yi=1 FIG. 11.— Biases in constraints on cosmological parameters from The first term represents the Poisson probability for having analyses of lens statistics. We show the errors ∆ΩM =ΩmMod- ΩtMrue and Nobs observedlenseswhenNpred arepredicted,whilethesec- ∆ΩΛ=ΩmΛodΩtΛruethatresultfromusingsimplemodelswithsphericallenses ond term represents the probability that the lenses have the thatneglectshiftsintheopticaldepth(crosses),imageseparations(triangles), orboth. (Seetextfordetails.) Thestatisticaluncertainties aresmallerthan observedproperties(e.g.,observedlensredshiftz andimage l thesizeofthepoints. separation ∆θ given the source redshift z ). As mentioned s above,weneglectellipticityandshearinthelikelihoodanal- ysisbecausewewanttounderstandthebiasesthatmayoccur whensphericalmodelsareusedforlensstatisticanalyses.We distribution can affect cosmological constraints. In the first holdthe parametersin the velocitydispersion functionfixed case, we imagine using spherical lens models but manually at their input values since uncertainties in these parameters adjustingtheopticaldepth.Thisisequivalenttochangingthe have negligible effect (Mitchelletal. 2004). Thus, the only totalnumberofdeflectors. Inpractice,itmeansadjustingthe variables in the model are the cosmological parameters Ω numberofsourcesinourmocksurvey(sincewefixthenum- M andΩΛ,whichweadjusttomaximizethelikelihood. Weuse ber of lenses). The crosses in Figure 11 show the errors in inputvaluesofΩM =0.3andΩΛ=0.7,andstudyhowmuch the recovered cosmological parameters if the difference be- the recoveredvalues differ. As mentioned above, using sur- tweentheactual(input)opticaldepthandthesphericalmodel veyswith 1000lensesshouldmitigatePoissonuncertainties, is (- 20,- 15,- 10,- 5,0,5,10,15,20)%. We see that simply butwealwaysproduceandanalyze10independentsurveysto changingthe optical depth moves the cosmologicalparame- verifythatthestatisticalnoiseinourresultsisnegligible. tersmainlyalongthelinecorrespondingtoflatcosmologies, Itis usefulto beginbyexaminingtwo toy modelsthatfo- and the shift is fairly small: ∆ΩΛ =0.03 if the real optical cusonhowchangesintheopticaldepthorimageseparation depthis10%largerthanpredictedbythesphericalmodel. 9 In the second case, we again start with spherical models wedonotexpectellipticityandsheartohavealargeeffecton but manually adjust the image separations. This is equiva- thepredictednumberoflensesinfuturelenssurveys. lent to shifting the velocity dispersion distribution to higher Ellipticityandsheardonotshiftthemeanofthedistribution or lower values, and then adjusting the number of galax- oflensimageseparations,buttheydointroduceanadditional ies to keep the optical depth fixed. The triangles in Fig- scatter of ∼10%. They naturallyaffect the relative numbers ure 11 show the results of shifting the image separations by ofdouble,quadruple,andtriplelenses,buttheycannoteasily (- 20,- 15,- 10,- 5,0,5,10,15,20)%. Thereis a largeshiftin explain the high observed quadruple-to-double ratio. Ellip- the recovered cosmological parameters, and it is almost or- ticityhaslittleeffectonpredictionsforelusivecentrallensed thogonalto the line of flat cosmologies. For example, if the images,althoughitdoesleadtoasegregationthatquadruple realimageseparationsare10%largerthanpredictedbyspher- lenses are generally expected to have fainter central images ical models, then there will be errors of ∆Ω = 0.12 and thandoublelenses(Keeton2003). M ∆ΩΛ =0.21 in the parameters recovered by spherical mod- Since ellipticity and shear produce only small changes in els. Thesetwocasesarejusttoyexamples,buttheyillustrate the lensing optical depth and image separation distribution, theimportantprinciplethatevensmallerrorsinthemodelim- theyare notveryimportantinlensingconstraintsoncosmo- ageseparationscanhaveasignificanteffectoncosmological logical parameters. Neglecting them leads to biases in Ω M constraints(evenifsmallerrorsintheopticaldepthdonot). andΩΛ of<0.02. Moreover,hydrodynamicalN-bodysimu- Finally,weconsiderthecasewhereweusethefulleffects lationstendtofindsystemsthataremoresphericalthanthose of ellipticity and shear on the mocksurvey. Essentially, this in dissipationless simulations (e.g., Kazantzidisetal. 2004). amounts to using the corrections to the optical depth from Therefore,theellipticityeffectsonlensingstatisticsfoundin Figure 5 and to the image separation distribution from Fig- thispaper,whilealreadysmall,couldevenbeanoverestimate. ure 8. The circle in Figure 11 shows that neglecting ellip- Weconcludethatforlensstatisticsproblemsotherthanim- ticity and shear in the likelihood analysis causes errors of agemultiplicities,ellipticityandshearhavesurprisinglylittle ∆ΩM =0.00±0.01 and ∆ΩΛ =- 0.02±0.01, where the er- effect. Unless percent-levelprecision is needed, or a survey rorbars represent the statistical uncertainties in our calcula- with a particularly steep LF is being considered, ellipticity tions. (We have achieved small Poisson uncertainties but and shear can probably be ignored. Their effects will be- not eliminated them altogether.) That is the case if we al- comemoreimportantaslenssamplesgrowintothehundreds low ΩM and ΩΛ to vary independently. If we restrict atten- or thousandsand statistical uncertaintiesplummet(see, e.g., tion to flat cosmologies (ΩM+ΩΛ =1) then the bias is just Kuhlenetal.2004). Atthattimeitwillbeimportanttoknow ∆ΩM =- ∆ΩΛ=0.01(withnegligibleerrorbars). Thisresult the distributions of ellipticity and shear, and also to resolve isconsistentwithourconclusionsfromtheprevioussections questionsabouthowtonormalizethelensmodels(see§2.2). thatellipticityandshearhavelittleeffectontheopticaldepth There are systematics besides ellipticity and shear that andmeanimageseparation.Itisnonethelessvaluabletohave may affect strong lens statistics. They include mergers and acarefulvalidationoftheconventionalwisdomthatellipticity evolution in the deflector population (e.g., Rixetal. 1994; andsheardonotsignificantlyaffectcosmologicalconstraints Mao&Kochanek 1994; Keeton 2002; Ofeketal. 2003; derivedfromlensstatistics. Chae&Mao 2003; Mitchelletal. 2004), halo triaxiality (e.g., Oguri&Keeton 2004) or other complex internal 7. CONCLUSIONS structure (e.g., Mölleretal. 2003; Quadrietal. 2003), Theeffectsofellipticityandshearonstronglensingstatis- compound lens galaxies (e.g., Kochanek&Apostolakis tics have been swept under the rug in most analyses to date 1988; Möller&Blain 2001; Cohn&Kochanek 2003), (a valiant exception being Chae 2003). The reason for this and lens galaxy environments (e.g., Mölleretal. 2002; is twofold: (1) models with nonspherical deflectors intro- Keeton&Zabludoff 2004). In order to bring lens statistics ducenew,andsometimespoorlyconstrained,parametersand into the realm of precision cosmology, each of these factors greatlycomplicatecalculations;and(2)conventionalwisdom mustbe addressed carefully. We havetaken one step in that suggestedthatrealisticellipticitiesandshearshadlittleeffect direction by studying ellipticity and shear, finding that their onanythingbuttheimagemultiplicities.Wehavesteppedbe- effectsare relatively small and in principle easy to take into yondthestateofblissfulignorancetopresentageneralanal- account. ysisofhowellipticityandshearenterintolensstatistics. The effects depend strongly on magnification bias, which in turn depends on the luminosity function of sources in a lenssurvey. IftheLFisa powerlaw ∝L- β, asinradiosur- WethankAndreyKravtsovforinterestingdiscussionsthat veys like CLASS (with β ≈2.1), ellipticity and shear gen- promptedus to examine the biases in cosmologicalparame- erally decrease or increase the lensing opticaldepth by only ters.WethankChrisKochanekandtheanonymousrefereefor a few percent. The increase is more than 5% only if the LF good questions about the model normalization. DH is sup- is steep (β &2.5). For optical quasar surveys, if the limit- ported by the DOE grant to CWRU. CRK is supported by ingluminosityisbelowthebreakinthequasarLFthenellip- NASAthroughHubbleFellowshipgrantHST-HF-01141.01- ticity and shear decrease the opticaldepth by a few percent. A from the Space Telescope Science Institute, which is op- Thereisanoticeable(>5%)increaseonlyforsurveyslimited eratedbytheAssociationofUniversitiesforResearchinAs- tothebrightestquasars(Lcut/L∗&10,ifthebrightendslopeis tronomy,Inc.,underNASAcontractNAS5-26555.C-PMais β =3.43).Sinceongoingandplannedopticalsurveysareex- supportedbyNASAgrantNAG5-12173andaCottrellSchol- h pectedtoreachtothefaintendoftheQSOLF(Lcut/L∗.1), arsAwardfromtheResearchCorporation. 10 APPENDIX WHY ISA/A ≤1? 0 In this appendix we derive the cross section for a generalized isothermal lens to explain the result from § 3 that ellipticity reducesthecrosssection. ThelenspotentialforageneralizedisothermalmodelhastheformΦ (r,φ)=rf(φ)where f(φ)isan iso arbitraryfunctionspecifyingtheangularshape. Considerexpandingthepotentialinangularmultipoles, ∞ Φ (r,φ)=θ r 1- a cos(mφ)+b sin(mφ) , (A1) iso E m m ! Xm=1h i whereθ istheEinsteinradius(asdefinedin§2.2). Thecorrespondingmassdistributionthenhastheform E ∞ θ κ (r,φ)= E 1+ ǫ cos[m(φ- φ )] , (A2) iso m m 2r ! m=1 X wheretheamplitudeǫ anddirectionφ ofthemassmultipolearegivenby m m ǫ =(m2- 1) a2 +b2 , (A3) m m m 1 qb φ = tan- 1 m . (A4) m m a m Inotherwords,wecanthinkofthismodelasamultipoleexpansionofthesurfacemassdensity. Theradialcaustic—properlytermedapseudo-causticsinceasingularisothermallensdoesnotformallyhavearadialcritical curve(seeEvans&Wilkinson1998)—canthenbewritteninparametricformas: ∞ u (λ)=- θ cosλ+ (a cosmλ+b sinmλ)cosλ+m(a sinmλ- b cosmλ)sinλ (A5) caus E m m m m ! Xm=1h i ∞ v (λ)=- θ sinλ+ (a cosmλ+b sinmλ)sinλ- m(a sinmλ- b cosmλ)cosλ (A6) caus E m m m m ! Xm=1h i Althoughthisformappearscomplicated,ifwe collectthetwocoordinatesu andv intoa vector~u thenwecaneasily caus caus caus evaluatetheareainsidetheradialcaustic: 2π 1 d~u 1 ∞ A= ~u (λ)× caus dλ=πθ2 1- (a2 +b2)(m2- 1) . (A7) 2 caus dλ E 2 m m Z0 (cid:20) (cid:21) " m=1 # X Thisisthelensingcrosssection(providedtherearenonakedcusps).Thesummand,andhencethesum,ismanifestlynonnegative, sothecrosssectionforanynonsphericalmodelisA<A ≡πθ2. ThisresultisillustratedinFigureA12fordifferentmultipole 0 E terms. Itisclearthatasphericitydeformsthecausticsinsuchawaythatthecrosssectionissmallerthanforthesphericalcase. 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