Draftversion February2,2008 PreprinttypesetusingLATEXstyleemulateapjv.08/13/06 THE SLOAN LENS ACS SURVEY. VI: DISCOVERY AND ANALYSIS OF A DOUBLE EINSTEIN RING†. Rapha¨el Gavazzi1, Tommaso Treu1,2,3, L´eon V. E. Koopmans4, Adam S. Bolton5,6, Leonidas A. Moustakas7, Scott Burles8, and Philip J. Marshall1 Draft versionFebruary 2, 2008 ABSTRACT We report the discovery of two concentric partial Einstein rings around the gravitational lens SDSSJ0946+1006, as part of the Sloan Lens ACS Survey. The main lens is at redshift z = 0.222, l while the inner ring (1) is at redshift z = 0.609 and Einstein radius R = 1.43 0.01′′. The 8 widerimageseparation(R =2.07 0s.102′′)oftheouterring(2)impliestEhina1titisath±igherredshift 0 Ein2 ± than Ring 1. Although no spectroscopic feature was detected in 9 hours of spectroscopy at the 0 ∼ 2 Keck I Telescope, the detection of Ring 2 in the F814W ACS filter implies an upper limit on the redshift of zs2 .6.9. The lens configuration can be well described by a power law total mass density n profile for the main lens ρ r−γ′ with logarithmic slope γ′ =2.00 0.03 (i.e. close to isothermal), Ja velocity dispersion σ =tot2∝87 5kms−1 (in good agreement with±the stellar velocity dispersion SIE 0 σv,∗ =284 24kms−1) with litt±le dependence upon cosmological parameters or the redshift of Ring ± 1 2. Using strong lensing constraints only we show that the enclosed mass to light ratio increases as a functionofradius,inconsistentwithmassfollowinglight. Adoptingaprioronthestellarmasstolight ] ratio from previous SLACS work we infer that 73 9% of the mass is in form of dark matter within h ± the cylinder ofradius equalto the effective radiusof the lens. We consider whether the double source p plane configurationcanbe used to constrain cosmologicalparameters exploiting the ratios of angular - o distance ratios entering the set of lens equations. We find that constraints for SDSSJ0946+1006 are r uninteresting due to the sub-optimal lens and source redshifts for this application. We then consider t s the perturbing effect of the mass associated with Ring 1 (modeled as a singular isothermal sphere) a building a double lens plane compound lens model. This introduces minorchangesto the mass of the [ main lens, allows to estimate the redshift of the Ring 2 (zs2 = 3.1+−21..00), and the mass of the source 1 responsible for Ring 1 (σ = 94+27kms−1). We conclude by examining the prospects of doing SIE,s1 −47 v cosmographywithasampleof50doublesourceplanegravitationallenses,expectedfromfuturespace 5 based surveys such as DUNE or JDEM. Taking full account of the uncertainty in the mass density 5 profile of the main lens, and of the effect of the perturber, and assuming known redshifts for both 5 sources,wefindthatsuchasamplecouldbeusedtomeasureΩ andw with10%accuracy,assuming 1 m a flat cosmologicalmodel. . 1 Subjectheadings: Gravitationallensing–galaxies: Ellipticalsandlenticulars,cD–galaxies: structure 0 – galaxies: halos – cosmology: dark matter – cosmology: cosmologicalparameters 8 0 v: 1. INTRODUCTION Measuringthemassdistributionofgalaxiesisessential for understanding a variety of astrophysical processes. i X Electronicaddress: [email protected] Extended mass profiles of galaxies provide evidence for r †Based on observations made with the NASA/ESA Hubble darkmattereitherusingrotationcurves(e.g.Rubinetal. a SpaceTelescope,obtainedattheSpaceTelescopeScienceInstitute, 1980;van Albada et al. 1985; Swaters et al. 2003), weak whichisoperatedbytheAssociationofUniversitiesforResearchin lensing (e.g. Brainerd et al. 1996; Hoekstra et al. 2004; Astronomy,Inc.,underNASAcontractNAS5-26555. Theseobser- vationsareassociatedwithprogram#10886. Supportforprogram Sheldon et al. 2004; Mandelbaum et al. 2006), or dy- #10886 was provided by NASA through a grant from the Space namics of satellite galaxies (e.g. Prada et al. 2003; Con- Telescope Science Institute, which is operated by the Association roy et al. 2007) which is one of the main ingredients of ofUniversitiesforResearchinAstronomy,Inc.,underNASAcon- the standard Λ cold dark matter (ΛCDM) cosmologi- tractNAS5-26555. 1Department of Physics, University of California, Broida Hall, cal model. At galactic and subgalactic scales, numerical SantaBarbara,CA93106-9530, USA cosmological simulations make quantitative predictions 2SloanFellow regarding, e.g., the inner slope of mass density profiles 3PackardFellow 4KapteynAstronomicalInstitute, UniversityofGroningen,PO and the existence of dark matter substructure. Precise box800,9700AVGroningen,TheNetherlands mass measurements are key to test the predictions and 5Institute for Astronomy, Universityof Hawaii, 2680 Wodlawn provide empirical input to further improve the models. Dr.,Honolulu,HI96822, USA Gravitational lensing has emerged in the last two 6Harvard-Smithsonian Center for Astrophysics, 60 Garden St. decades as one of the most powerful ways to measure MS-20,Cambridge,MA02138, USA 7Jet Propulsion Laboratory, Caltech, MS 169-327, 4800 Oak the mass distributions of galaxies, by itself or in com- GroveDr.,Pasadena, CA91109, USA bination with other diagnostics. Although strong grav- 8DepartmentofPhysicsandKavliInstituteforAstrophysicsand itational lenses are relatively rare in the sky (. 20 per Space Research, Massachusetts Institute of Technology, 77 Mas- squaredegreeatspace-baseddepthandresolution;Mar- sachusetts Ave.,Cambridge,MA02139,USA shall et al. 2005; Moustakas et al. 2007), the number 2 Gavazzi et al. of known galaxy-scale gravitational lens systems has in- The goal of this paper is to study and model this pe- creasedwellbeyondahundredasaresultofanumberof culiar system in detail, as an illustration of some astro- dedicated efforts exploiting a variety of techniques (e.g. physical applications of double source plane compound Warren et al. 1996; Ratnatunga et al. 1999; Kochanek lenses,includingi)thedeterminationofthemassdensity et al. 1999; Myers et al. 2003; Bolton et al. 2004; Ca- profile of the lens galaxy independent of dynamical con- banac et al. 2007). The increased number of systems, straints; ii) placing limits on the mass of source 1 based together with the improvement of modeling techniques on multiple lens plane modeling; iii) estimating the red- (e.g. Kochanek & Narayan 1992; Warren & Dye 2003; shift of source 2 and the cosmological parameters from Treu & Koopmans 2004; Brewer & Lewis 2006; Suyu the angular distance size ratios. The paper is therefore et al. 2006; Wayth & Webster 2006; Barnab`e & Koop- organized as follows. Section 2 summarizes the observa- mans 2007), has not only enabled considerable progress tions,photometricandspectroscopicmeasurements,and in the use of this diagnostic for the study of the mass discusses the morphology of the lens system. Section distribution of early and most recently late-type galax- 3 describes our gravitational lens modeling methodol- ies, but also for cosmography, i.e. the determination of ogy. Section 4 gives the main results in terms of con- cosmological parameters (e.g. Golse et al. 2002; Soucail straints on the mass distribution of the lens galaxy and et al. 2004; Dalal et al. 2005). of source 1. Section 5 discusses the use of double source Given the already small optical depth for strong lens- planelensesasatoolforcosmographyusingtheexample ing, the lensing of multiple background sources by a of SDSSJ0946+1006 and also addresses the potential of single foreground galaxy is an extremely rare event. large samples of such double source plane lenses for the At Hubble Space Telescope (HST) resolution (FWHM samepurpose. Insection6wesummarizeourresultsand 0′.′12) and depth (I 27) it is expected that one briefly conclude. AB ∼ ∼ massive early-type galaxy (which dominate the lensing Unless otherwise stated we assume a concordance cos- cross-section) in about 200 is a strong lens (Marshall mology with H = 70h kms−1Mpc−1, Ω = 0.3 and 0 70 m et al. 2005). Taking into account the strong dependence Ω = 0.7. All magnitudes are expressed in the AB sys- Λ of the lensing cross-section on lens galaxy velocity dis- tem. persion ( σ4), and the population of lens galaxies, we ∝ 2. DATA estimate that about one lens galaxy in 40 80 could be a double source plane strong gravita∼tiona−l lens (see The lens galaxy SDSSJ0946+1006 was first identified appendix A). For these reasons, at most a handful of inthespectroscopicSDSSdatabasebasedontheredshift doublelensesaretobefoundinthelargestspectroscopic ofthelensinggalaxyzl =0.222andthatofabackground surveys of early-type galaxies such as the luminous red source at zs1 = 0.609 (hereafter source 1), as described galaxiesoftheSloanDigitalSkySurvey. However,future by Bolton et al. (2004, 2006a), and Bolton et al. (2008, highresolutionimagingsurveyssuchasthoseplannedfor in prep.). This section describes HST follow-up imaging JDEM and DUNE (Aldering & the SNAP collaboration ( 2.1), the properties of the lens ( 2.2)and lensed ( 2.3) § § § 2004; R´efr´egier et al. 2006) will increase the number of galaxies. knownlensesby2-3ordersofmagnitude(Marshalletal. 2.1. Hubble Space Telescope observations & data 2005), and hence should be able to provide large statis- reduction ticalsamples ofdouble sourceplane gravitationallenses, opening up the possibility of qualitatively new applica- SDSSJ0946+1006 was then imaged with the ACS on tions of gravitationallensing for the study of galaxy for- board the HST (cycle 15, Prog. 10886, PI Bolton). mation and cosmography. The Wide Field Channel with filter F814W was used We reporthere the discoveryofthe firstdouble source for a total exposure time of 2096 s. Four sub- plane partial Einstein Ring. The gravitational lens sys- exposures were obtained with a semi-integer pixel offset tem SDSSJ0946+1006 , was discovered as part of the (acs-wfc-dither-box) to ensure proper cosmic ray re- Sloan Lens ACS (SLACS) Survey (Bolton et al. 2005, moval and sampling of the point spread function. The 2006a; Treu et al. 2006; Koopmans et al. 2006; Bolton image reduction process is described in (Gavazzi et al. etal. 2007;Gavazziet al.2007). The objectwas firstse- 2007) and results in a 0′.′03/pixel spatial sampling. This lectedbythepresenceofmultipleemissionlinesathigher pixel size provides good sampling of the PSF for weak redshift in the residuals of an absorption line spectrum lensingapplications,atthe(small)priceofinducingnoise from the SDSS database as described by Bolton et al. correlationoverscalesof1-2pixels. Thisisaccountedfor (2004) and then confirmed as a strong lens by high res- inouranalysisbycorrectingpixelvariancesaccordingto olution imaging with the Advanced Camera for Surveys the procedure described by Casertano et al. (2000). aboard HST. In addition to an Einstein ring due to the Figure 1 shows the HST image of the lens galaxy field source (hereafter source 1) responsible for the emission together with an enlarged view of the lensed features, lines detected in the SDSS spectrum, the Hubble image after subtraction of a smooth model for the lens surface also shows a second multiply imaged system forming a brightness distribution. For reference, one arcsecond in brokenEinsteinRingwithalargerdiameterthenthe in- thelensplanesubtendsaphysicalscaleof3.580h−1kpc. 70 nerring(hereaftersource2). Thisconfigurationcanonly arise if the two lensed systems are at different redshifts 2.2. Lens galaxy properties andwellalignedwiththe centerofthe lensinggalaxy. It The two-dimensional lens surface brightness was fit- isagreatopportunitythatadoublesourceplanelenshas ted with galfit (Peng et al. 2002) using two elliptical been found among the approximately 90 lenses discov- S´ersiccomponents. The additionofa secondcomponent ered by the SLACS collaboration to date (Bolton et al. isneededtoprovideagoodfitinthecenter,andtorepro- 2008, in prep.). duce the isophotal twist in the outer regions. To reduce The double Einstein ring SDSSJ0946+1006 3 Fig. 1.— HST/F814W overview of the lens system SDSSJ0946+1006. The right panel is a zoom onto the lens showing two concentric partial ring-like structures after subtracting the lens surface brightness. the effect of lensed features in the fit we proceeded it- et al. (2008, in prep.) who considered de Vaucouleurs eratively. We first masked the lensed features manually, surface brightness distributions (n 4 by construction). ≡ then we performed galfit fits creating masks by 4-σ Notethatweusethesameconventionforallcharacteris- clipping. Two iterations were needed to achieve conver- ticradiireportedthroughout. Forellipticaldistributions gence. radii are expressed at the intermediate radius (i.e. the The total magnitude of the lens obtained by sum- geometric mean radius r =√ab). ming the flux of the two S´ersic models is F814W = Inaddition, the stellarvelocitydispersionσ =263 ap 17.110 ± 0.002 after correction for Galactic extinction 21kms−1 was measured with SDSS spectroscopy withi±n (Schlegel et al. 1998). The rest-frame V band abso- a 3′′ diameter fiber. We convert this velocity dispersion lute magnitude is M = 22.286 0.025 using the K- correction of Treu etVal. −(2006). ±The errors are dom- σap into the fiducialvelocityσv,∗ thatenters Fundamen- tal Plane analyses and measured in an aperture of size inated by systematic uncertainties on the K-correction R /8usingtherelationσ /σ =(R /8/R )−0.04 term. The most concentratedS´ersic component c dom- eff v,∗ ap eff ap ≃ 1 1.08 (see Treu et al. 2006, and references therein) inates at the center and accounts for about 17.5%of the Basedonphotometric redshifts availableonline on the total lens flux. The effective radius of c is about 0.4′′ 1 SDSS webpage (Oyaizu et al. 2007), we note that the whereas that of c is 3′′ with about 10% relative ac- 2 ∼ lens galaxy is the brightest galaxy in its neighborhood. curacy. Similarly, the S´ersic indexes are n 1.23 and c1 ≃ Another bright galaxy about 40 arcsec south-west of n 1.75. c2 ≃ SDSSJ0946+1006 exhibits perturbed isophotes (an ex- To measure the one dimensional light profile of the tended plume) suggesting that it may have flown by re- lens galaxy, we used the IRAF task ellipse. Fig. 2 cently and might end up merging onto the lens galaxy. shows the radial change of ellipticity and position angle Its photometric redshift is z =0.20 0.04 consistent of the light distribution. There is a clear indication of phot ± with SDSSJ0946+1006 redshift. The extended envelope a sharp change in position angle and ellipticity between captured by the double S´ersic component fit also sup- 1 2′′. Thisisophotaltwistiswellcapturedbythedouble − ports the recent flyby hypothesis (e.g. Bell et al. 2006). S´ersic profile fit, that requires different PAs for the two components. Thereforeweconclude thatthe lens galaxy 2.3. Lensed structures is made of two misaligned components, having similar surface brightness at radius 0.6′′. Two concentric partial ring-like structures are clearly For comparison, a single ∼component S´ersic fit yields seen at radii1.43 0.01′′ and 2.07 0.02′′ from the cen- ± ± n 3.73, consistent with the typical light profiles of terofthe lens galaxy(Figure 1). Sucha peculiarlensing ma≃ssive early-type galaxies. The effective radius of the configuration–withwidelydifferentimageseparationsof composite surface brightness distribution is found to be nearly concentric multiple image systems – implies that R = 2.02 0.10 arcsec 7.29 0.37h−1kpc, where theringscomefromtwosourcesatdifferentredshift,the weeffassumed±a typical relat≃ive unc±ertainty70of about 5% innermost (Ring 1) corresponding to the nearest back- as discussed in (Treu et al. 2006). It is also consistent ground source 1 and the outermost (Ring 2) being sig- with an independent measurement reported by Bolton nificantly further away along the optical axis. Ring 1 has a typical cusp configuration with 3 merg- 4 Gavazzi et al. 2 is detected in the ACS/F814W filter, we can set an upper limit on its redshift z < 6.9 by requiring that s2 the Lymanbreakbe atshorterwavelengthsthanthe red cutoff of the filter. 3. LENSMODELING 3.1. Model definition This section describes our adopted strategy to model thisexceptionallenssystem. Webeginwithasimplifying assumption. Although the gravitational potential arises frombothastellarandadarkmattercomponent,asingle powerlawmodelforthetotaldensityprofileturnsoutto be agooddescriptionofSLACSlenses (Koopmansetal. 2006). Therefore, we assume the total convergence for a source at redshift z to be of the form: s κ(~r,z )= bγ∞′−1 x2+y2/q2 (1−γ′)/2 Dls , (1) s 2 D (cid:0) (cid:1) os with 4 free parameters: the overall normalization b, the Fig. 2.—ResultsforisophotalfitwithIRAF/ellipse. Top logarithmic slope of the density profile γ′, the axis ra- panel: Position angle versus radius. Middle panel: Axis tio q and position angle PA (omitted in Eq. (1) for 0 ratioversusradius. Theverticallinesshowthelocationofthe simplicity) of iso-κ ellipses. The familiar case of the innerandouterEinsteinringswhichweremaskedoutduring singular isothermal sphere is that corresponding to a the fitting process. We also overlay in the top and middle slope γ′ = 2 and q = 1. In this case b relates ∞ panels as a blue solid line the ellipse output performed on to the velocity dispersion of the isothermal profile by the best fit galfit two-dimensional brightness distribution. b = 4π(σ /c)2 = (σ /186.2kms−1)2 arcsec. Note Bottom panel: best fit S´ersic profiles obtained with galfit. ∞ SIE SIE that σ is nothing but a way of redefining the normal- The formal error bars on the surface brightness profile are SIE smaller than thedata points. ization of the convergenceprofile and does not necessar- ily correspond in a straightforward sense to the velocity ingconjugateimagesandacounterimageontheopposite dispersion of stars in the lens galaxy. In general, for ev- side of the lens and closer to the center than the large erycombinationofmodelparameters,thestellarvelocity cusp “arc”. This constrains the orientation of the lens dispersionofaspecifiedtracerembeddedinthepotential potential major axis to pass almost through the middle can be computed by solving the Jeans equation and will ofboth arcs. Ring 1 is among the brightestones to have be a function of radius and observational effects such as been discovered in the SLACS survey (See Bolton et al. aperture and seeing. 2008,in prep., for the latest compilation). The observed No assumptions are made aboutthe orientationof the F814Wmagnitudeism1 =19.784 0.006(extinctioncor- position angle PA0 of the lens potential. In addition, rected). Theerrorbarincludesonl±ystatisticaluncertain- we allow for external shear with modulus γext and po- ties. An additional systematic error of order . 0.1 mag sition angle PAext. For a multiple source plane system, is likely present due to uncertainties in the lens galaxy it is necessary to define a lens plane from which the ex- subtraction (Marshall et al. 2007). ternal shear comes from since shear has to be scaled by Ring 2 presents a nearly symmetrical Einstein cross the apropriate Dls/Dos term for each source plane. For configuration(withafaintbridgebetweenthe northand simplicity we assume that the global effect of external westimages),implyingthatthesourcemustlieveryclose pertubations comesfromthe same lens plane zl =0.222. to the optical axis. The observed F814W magnitude is We expectastrongdegeneracybetweeninternalelliptic- m = 23.68 0.09, making it about 36 times fainter ity and external shear but include this extra degree of 2 than Ring 1.±As for Ring 1 the error bar includes only freedom in the model to account for any putative twist statistical uncertainties. of isopotentials, as suggested by the observed isophotal NoevidenceofRing2ispresentintheSDSSspectrum. twist in the lens galaxy surface brightness. Note also This can be explained by the low peak surface bright- thatthe needofbeing able to handletwo distinctsource nessofRing2( 23mag arcsec−2)andlessimportantly planes led us to the somewhat unusual definition of b∞ by the diameter∼of the second ring being slighty larger in Eq. (1). With this convention, (b∞√q)γ′−1Dls/Dos thanthe3′′SDSSfiber(althoughseeBoltonetal.2006b, is the quantity closest to the b (or R ) parame- SIE Einst for a successful redshift measurement in a similar case). ter used in other SLACS papers (Koopmans et al. 2006; Deeper longslit spectroscopy was obtained at Keck Ob- Bolton et al. 2008). Note also that the center of mass is servatorywith the Low Resolution Imager Spectrograph assumedtomatchexactlythelensgalaxycenteroflight. (LRIS) instrument on December 22-23 2006, the total Theunknownredshiftofsource2isalsotreatedasafree integrationtime beingabout9hours. The goalwastwo- parameter, for which we assign a uniform 1 z 6.9 s2 ≤ ≤ fold: i) obtain the redshift of Ring 2; ii) measure the prior. Altogether,weuse7freeparameterstocharacter- stellar velocity dispersion profile of the main lens. This ize the potential of SDSSJ0946+1006: b , γ′, q, PA , ∞ 0 latter aspect will be presented elsewhere. Despite the γ , PA and z . ext ext s2 large integration time, we could not measure the source In this section and the next, we neglect the extra fo- redshift z due to a lack of emission lines in the range cusingeffectofRing1actingasalensonRing2,leaving s2 [3500, 8600˚A] that do not belong to Ring 1. Since Ring the discussion of this perturber for Section 5. The double Einstein ring SDSSJ0946+1006 5 3.2. Methods 4 spots identified in Ring 1, two of them having 3 clear conjugations(S1a, S1c)whereasthe othertwohaveonly We considerthree strategiesfor studying gravitational have2(S1b,S1d). OnesinglebrightspotinRing2isim- lens systems with spatially resolved multiple images. aged4times. Thetypicalrmserrormadeonthelocation The first one consists of identifying conjugate bright of spots estimated to be 0.03′′. Table 1 summarizes the spots in the multiple images and minimizing the dis- coordinates of matched points in the same frame as Fig. tance of conjugate points in the source plane. This ap- 1. For each knot S1a, S1b, S1c, S1d and S2, multiple proach is statistically conservative in the sense that it images with positive parity have an odd labelling num- only takes partial advantage of the large amount of in- ber. To guide the fitting procedure we also demand the formationpresentinthedeepHSTdata. However,ithas image parity to be preserved by the model. Therefore, the benefit of being robust and relatively insensitive to taking into account the unknown position of these spots the details ofthe sourcemorphology,andotherconcerns inthesourceplane,weenduphaving18constraints(see that affect different alternative techniques in the case. Gavazziet al. 2003)whereas the consideredmodel has 7 Thesecondapproachisthelinearsourceinversionand free parameters. Hence the optimizationproblemhas 11 parametric potential fitting method described by War- degrees of freedom. ren&Dye (2003),Treu&Koopmans(2004),Koopmans (2005)andSuyuetal.(2006). Astrongadvantageofthis method is that it takes fully into account the amount TABLE 1 of information contained in each pixel. Although this Summaryof pixelcoordinates used forlens modeling. method is robust, there are many degrees of freedom to Img. 1 Img. 2 Img. 3 Img. 4 modeltheintrinsicsourcesurfacebrightnessdistribution S1a 0.34,-1.50 -0.94,0.68 1.52,0.19 – and thus some form of regularisation is needed to avoid S1b – -1.16,0.22 1.44,0.88 – fitting the noise as described in the references above. S1c -0.43,-1.42 -1.10,0.67 1.23,0.88 – The third method (e.g. Marshall et al. 2007; Bolton S1d -0.14,-1.68 -0.57,0.96 – – et al. 2007, 2008) describes the source as one or several S2 -1.51,-1.78 1.56,-1.19 1.55,1.65 -1.34,1.32 componentsparameterizedwithellipticalsurfacebright- ness profiles (usually S´ersic). In general, this method Positions (x,y) of each multiple knot are expressed in arcsec provides good fits to the data as long as not too many (typical rms error 0.03′′) relative to the lens galaxy surface suchcomponentsareneededtorepresentthesource,and brightness peak (got from galfit modeling, see §2.2). The frame position angle is 161.348◦ relative toNorth direction. directly provides physically meaningful parameters for the source. For high signal-to-noiseratio images of com- plex lensed features the dimensionality of the problem 4. MODELINGRESULTS may increase very fast. The optimization process and the exploration of the Inthecaseofamultiplesourceplanesystem,twodiffi- parameter space were performed by sampling the poste- cultiesarisewhenusingthesecondandthirdtechniques. rior probability distribution function with Monte-Carlo 1) Our current pixellized method does not handle mul- MarkovChains(MCMC).We assumedflatpriors. Table tiple source planes (see e.g. Dye et al. 2007, for recent 2 summarizes the results (“best fit” values are defined progress along this line). 2) The statistical weight given as the median value of the marginalizedPDF) and their to each of the partial rings depends essentially on their corresponding 68% CL uncertainties after marginalizing relative brightness. Since Ring 1 is 36 times brighter the posterior over all the other parameters. The best fit than Ring 2, it completely dominates the fit. This has model yields a χ2/dof = 13.2/11 1.20 which is statis- the unwanted side effect that a physically uninteresting tically reasonable10. ≃ morphologicalmismatchofthe innerring,due forexam- pleto poormodelingofthe sourceorofthepointspread function,overwhelmsanymismatchinthephysicallyim- TABLE 2 portant image separation of the outer ring. Best-fit modelparametersforSDSSJ0946+1006using a singlelens plane. The goal of the present analysis is to confirm that SDSSJ0946+1006 is the first example of a galaxy-scale b∞ [arcsec] 2.54±0.09 double source plane system and illustrate what kind of γ′ 2.00±0.03 information can be inferred from such a configuration. axisratioq 0.869+0.017 −0.013 After experimenting with all three techniques – and in PA0 −11.8+−78..09 light of the difficulties described above – we decided to γext 0.067+−00..001007 focusonthemorestraightforwardconjugatepointsmod- PAext −31.5+−64..98 eling technique, using the other techniques to aid in our zs2 5.30+−11..0030 modeling. σSIE [kms−1] 287.0+−55..13 In practice, the modeling technique adopted here is “unlensed”apparentF814s1 [mag] 22.76±0.02±0.10 similar to the one used by Gavazzi et al. (2003). The “unlensed”absoluteVs1 [mag] −19.79±0.05±0.10 mergingcuspnatureofring1makesthe identificationof “unlensed”apparentF814s2 [mag] 27.01±0.09±0.10 quadruply imaged spots hazardous along the elongated Best fit model parameters and 68.4% confidence limits. Er- arcbutidentificationsaremucheasierbetweentheoppo- rors on magnitudes distinguish statistical (first) and system- site counter-image and the elongated arc. The identifi- atic from lens light subtraction (second). Angles are in de- cationofthebrightnesspeakS2inRing2isobvious. To grees oriented from North toEast. guide the identification process, we also used fits based on the pixellized source inversion. We ended up having 10 A χ2 distribution with 11 degrees of freedom gives a proba- bilityof28%thattheχ2 valuewillbegreaterthan13.2. 6 Gavazzi et al. Fig. 3.— Best-fit single lens plane model for the lens SDSSJ0946+1006. The model parameters were found using the identification of conjugate bright knots but the quality of the model is illustrated with a pixelised source inversion technique. Top left: observation with thelenslightprofilesubtractedoff. Top middle: modelprediction intheimage planeandassociated residuals (Top right). The model also predicts the light distribution in the source planes zs1 and zs2 (Bottom left and right respectively). Note a different color stretching for source plane 2 (factor 6) in this latter case. Critical and caustic lines corresponding to the two source planes are overlaid (smaller blue for zs1 =0.609 and wider red for zs2 =5). Theresultsofthebestfitmodelinferredfromthecon- jugationofbrightspotsisshowninFig. 3whereweused the pixellizedsourceinversiontechnique to illustrate the quality of the fit and the reliability of the conjugation method. Although the surface brightness of Ring 1 and Ring 2 identified by separate annuli in the image plane are inverted separately, model predictions in the image plane are recombined for convenience. The two source planes z =0.609 and z 5 are also shown. s1 s2 ≃ As expected, there is a degeneracy between external shearandellipticityofthetotalmassdistributionandthe modeling,suggestingthatthemajoraxisofthepotential and the external shear differ by PA PA = 20+12 0 − ext −16 deg,thatisthey arealignedwithin 1.2σ. The orienta- ∼ tion of external shear in agreement with the orientation of stars out to r .1′′ which is about 36 deg. The ori- − entation of the internal quadruple (lens ellipticity) and that of stars are misaligned by 24◦. Likewise,the axis Fig. 4.— Top panel: 68.3%, 95.4% and 99.3% CL con- ∼ tours for model parameters slope of the density profile γ′ ratio of the light distribution over this radial range is 0.85.b/a.0.93,again consistent with our lens model. aslnodpesoγu′racned2threedlsehnifstezqsu2i.valentBvoettloomcitypadniespl:erIsdioenm(dfoerfinthede thTehreedlesnhsiftmoofdesloinugrcaels2o:pzuts i=nte5r.e3stin1g.0co.nsTtrhaeinatcscoun- as 186.2pb∞q1/2/1′′kms−1). s2 ± racy is relatively low because of the saturation of the welllocalizedconstraintsonthe normalizationandslope Dls/Dos(zs) curve when zs . The top panel of Fig. of the radial total density profile. The lower panel of → ∞ 4 shows a mild correlation between zs2 and the slope of Fig. 4 shows the confidence regions for the slope γ′ and the density profile γ′. This is expected since the steeper the equivalent velocity dispersion σ . First, we find a SIE the density profile that fits the inner ring, the less mass totaldensityprofileveryclosetoisothermalwithaslope is enclosed between the two rings, and hence the further γ′ =2.00 0.03. The correspondingSIE velocitydisper- away must be the outer source. sion is σ ± = 287.0 5.2kms−1. In order to compare SIE In spite of the complexity of the azimuthal properties these resultswith SD±SS spectroscopy,one needs to solve of the lens potential, our modeling yielded stable and thesphericalJeansequationtakingintoaccountobserva- The double Einstein ring SDSSJ0946+1006 7 tional effects (SDSS fiber aperture, seeing) and the sur- We now compare these values to the stellar mass con- face density of dynamical tracers (radial distribution of tent in the effective radius using a the typical mass- stars in the lens galaxy) measured in 2.2. Here we as- to-light ratio of stellar populations in massive galax- § sume an isotropic pressure tensor. A generaldescription ies at that redshift M /L 3.14 0.32h (M/L ) ∗ V ≃ ± 70 V ⊙ of the method can be found in Koopmans (2006). Fig. (Gavazzi et al. 2007) and 30% intrinsic scatter about 5 shows the aperture velocity dispersion that would be this values (due to e.g. ag∼e-metallicity effects) as found measured with SDSS spectroscopic fibers when the den- in the local Universe (Gerhardet al. 2001;Trujillo et al. sity profile is normalized to fit the first ring alone. It 2004). This leads to a fraction of projected mass in the shows that slopes close to isothermal (γ′ 2) predict formofdarkmatterwithintheeffectiveradiusf (< ≃ DM,2D velocity dispersions close to SDSS spectroscopic veloc- R ) 73 9%whichisabouttwiceashighastheaver- eff ity dispersion, which gives strong support to our double age va≃lue f±ound by Gavazzi et al. (2007) and Koopmans source plane lensing-only analysis. Such a similarity is et al. (2006) thus making SDSSJ0946+1006 a particu- consistent with the results of previous SLACS studies larly dark-matter-richsystem. (Treu et al. 2006; Koopmans et al. 2006). We note that the accuracy reached on both the slope and the veloc- 5. EXPLOITINGTHEDOUBLESOURCEPLANE:BEYOND THELENSMASSPROPERTIES ity dispersion based on lensing constraints alone is bet- ter than that afforded by kinematical measurements at In this section we address two particular applica- thesameredshift,althoughthetwomethodsarecomple- tions afforded by the double source plane nature of mentary in their systematic errorsand degeneracies(see SDSSJ0946+1006. First, in 5.1 we discuss whether this § discussion in e.g. Treu & Koopmans 2002). particularsystemgivesinterestingconstraintsoncosmo- logical parameters. Then, in 5.2, we present a com- § pound double lens plane mass model and use it to con- strainthe total mass of the Ring 1. This provides a new (andperhaps unique)wayto obtaintotalmassesofsuch compact and faint objects. Thus, in combination with the magnifying power of the main lens, this application appearstobeapromisingwaytoshedlightonthenature offaintbluecompactgalaxies(e.g.Marshalletal.2007). In 5.3 we discuss the prospects of doing cosmography § with samples of double source plane lenses, taking into accountthe lensing effects of the inner ring on the outer ring. 5.1. An ideal optical bench for cosmography? Can a double source plane lens be used to constrain global cosmological parameters like Ω or Ω ? In prin- m Λ ciple this canbe done because lensing efficiency depends on the ratio of angular diameter distances to the source Fig. 5.—Predicted stellar aperture velocity dispersion σap D andbetweenthelensandthesourceD aswellasthe asitwould bemeasured withSDSSspectroscopic settingsas os ls projectedsurfacemassdensityΣ(~θ)inthelensplane. In a function of the slope of the density profile. The normal- ization of density profile is fixed to be consistent with the formulae, writing the lens potential experienced by light Einstein radius of Ring 1. The shaded area is the 1σ SDSS rays coming from a source plane as redshift zs as: measurement uncertainty. It shows a remarkable agreement between the double source plane analysis and the coupling 4GD D of kinematical + source 1 plane data, both favoring nearly ψ(~θ,zs)= c2 Dol ls Z d2θ′Σ(~θ′)ln|~θ−θ~′| (2) isothermal slopes. Note that σap and σSIE do not need to be os identical. ψ (~θ)Dls , (3) 0 ≡ D os 4.1. Budget of mass and light in SDSSJ0946+1006 and considering two images at positions ~θ and ~θ 1 2 The tight constraints on the projected mass profile coming from source planes at redshift zs1 and zs2, between the two Einstein radii can be compared to one can measure the ratio of distance ratios η ≡ tthhee tliogthatl dpirsotjreicbtuetdionVibnafenrdredmainss-§t2o.-2li.ghtInraptiaortwiciutlhairn, (tDhelsm/Dulotsi)pzlse2/i(mDalgs/eDs,ogsi)vzes1ndaisrseuctmlyptfrioonmstohnetphreoppeortteinestioafl the effective radius Reff ≃ 7.29h−701kpc is M/LV = ψ0(~θ) and its derivatives defining the deflection, conver- 11.54±0.51h70 (M/LV)⊙ (correspondingtoatotalpro- gence and shear at the positions of the images. jected mass 4.90 0.13 1011h−1M ). The logarith- Applicationsofthismethodtoclustersofgalaxieswith mic slope of the±project×ed enclo7s0ed ⊙total mass profile several multiply imaged systems at different source red- is 3 γ′ = 1.00 0.03, while the slope of the cumu- shifts – assuming simple parametric models for the clus- lativ−e luminosity ±profile close to the effective radius is ters – seem to favor Ωm < 0.5 cosmologies (Golse et al. dlogL(< r)/dlogr = 0.62 with much smaller uncer- 2002;Soucail et al. 2004). However,unknown systemat- tainty. Thereforetheprojectedmass-to-lightratioprofile ics lurk under the cluster substructure, which can intro- increases with radius as r0.38±0.03 aroundR with high ducesignificantlocalperturbationsofψ (~θ). Inprinciple eff 0 statistical significance. – at least judging qualitatively from the smoothness of 8 Gavazzi et al. the isophotes, and the smoothness of galaxy scale Ein- steinRings–onecouldhopethatmassiveellipticalgalax- iesbelesspronetothissortofsystematicbecausesource size is large comparedto the substructure angular scale. Intheprevioussectionweconstrainedz forthegiven s2 ΛCDM concordance cosmology. Here we re-parametrize the problem using η itself as a free parameter to con- strain the change in lensing efficiency between the two source planes. The left panel of Fig. 6 shows the joint constraints on the two parameters γ′ and η. A first con- sequence of this more general parameterization is that, by allowing a broader range of values for η (i.e. allow- ing more freedom in the cosmological model), the un- certainties on the slope are significantly increased: we find γ′ = 2.07 0.06. Steeper density profiles are now ± somewhatcompensated by arelatively higherlensing ef- ficiency for the second source plane. In other words, the Fig. 7.—68.3%, 95.4% and 99.3% CL contours in the red- tight constraints previously obtained on the slope of the density profile depend to some extent on the assumed shiftofsource2andΩmparameterspaceassuminganisother- maldensityprofile. Thisshows thatevenusingstrongpriors cosmologicalmodel(i.e. assumingΛCDMcosmologyled on the density profile and for a given source redshift, only to γ′ =2.00 0.03). loose constraints can be inferred on cosmological parameters ± with a single double source plane system. modeling strategy based on the identification of conju- gate knots underestimates the potential accuracy of the method. Statistical errors would decrease by a factor of a few with a full modeling of the surface brightness distribution in the image plane. Unfortunately, the er- ror budget would then be limited by additional system- aticsourcesofuncertaintylikeextraconvergencecoming from large scale structures along the line of sight with estimated standard deviation σ & 0.02 (Dalal et al. κ 2005) or due to a non trivial environment in the main lens plane. Therefore, we conclude that it is unlikely that any cosmographic test based on the unique multi- ple Einstein ring system SDSSJ0946+1006 will provide valuable information on cosmological parameters. The prospects of using large numbers of double source plane lenses are investigated in 5.3. Fig. 6.— Left panel: Constraints on the logarithmic slope § γ′ andtheratioof distanceratios η. Contoursenclose68.3% 5.2. Source 1, alias Lens 2 and 95.4% of probability. The right panel shows η(zs2) as a Among the massive perturbers along the line of sight function of zs2 for two flat cosmologies (Ωm,ΩΛ)=(0.3,0.7) (black) and (Ωm,ΩΛ) = (1.,0.) (red) which are two sensible tosource2,themostprominentisprobablythemassas- “extreme” cases. The dotted horizontal lines illustrate the sociated with source 1. Since the lens modeling predicts upperlimits onη forthesecosmologies giventheassumption that both sources are located very close to the optical zs2 ≤6.9 (see §2.3). axis (the center of the lens, see lower panels of Fig. 3), the light rays coming from source 2 to the observer will The right panel of Fig. 6 shows η(z ) as a function experience the potential of source 1 before that of the s2 of the second source redshift for two “extreme” flat cos- main lens. Fig. 8 illustrates the complexity of the con- mologies: (Ω ,Ω ) = (0.3,0.7) and (Ω ,Ω ) = (1.,0.), figuration which adds some extra positive focusing for m Λ m Λ intermediate cases lying in between. This shows that the second source plane. For the cosmological applica- high values η & 1.57 are not consistent with currently tions we mentioned above, this translates into a small favored cosmologies. The upper limit on η(z = 6.9) is but systematic source of bias. The bias introduced on s2 alsoshownforthese twocases. Thisillustratesthatvery the inferred mass profile of the main lens is small, so looseconstraintscanbeobtainedoncosmologicalparam- thatthe conclusions presentedin 4 arenot significantly § eters even if z were known spectroscopically. Likewise, altered except for the estimate of z . s2 s2 even assuming an isothermal slope of the density pro- Onthe bright side, this lens configurationallowsus to fileasmotivatedbyjointlensinganddynamicalanalyses obtain some insight on the mass associated with Ring (Koopmanset al.2006)does notdrasticallyimprovethe 1 (also identified as “Lens 2”) provided we now fully constraintsonηandconsequentlyoncosmologyasshown take into account the multiple lens plane nature of such inFig7,evenifz couldbemeasuredwithspectroscopic lines of sight(e.g. Blandford& Narayan1986;Schneider s2 precision. etal.1992;Bartelmann2003). Thisisthepurposeofthe However, it is important to point out that the formal present section, in which we fix the ΛCDM concordance 3% relative uncertainty we get on η from our lens cosmologicalmodel for simplicity. ∼ The double Einstein ring SDSSJ0946+1006 9 Fig. 8.— Sketch of the lensing optical bench with source 1 acting as a perturbing lens on source 2 which complicates the relation between redshifts, deflection angles and angular distances. Toachievethisgoalwehavetoaddressthemassprop- erties of the main lens at the same time as those of the firstsource1. Wereconsiderthelensmodelof 3,butadd § another mass component at redshift z =z =0.609 in l2 s1 the formof a singular isothermalsphere with free equiv- Fig. 9.— Left panel: contours in parameter space of alent velocity dispersion parameter σ and centered SIE,s1 the velocity dispersion of the main lens σSIE and that of on the position of source 1. As in 3, our uncertainty § the first source σSIE,s1. Given the tight correlation σSIE ≃ on the distance to source 2 is simply parameterized by (687−200.3γ′)kms−1 foundin §4, theupperabscissa shows its redshift zs2 in the context of a ΛCDM cosmological thecorrespondance with slope γ′. The kinematical SDSS es- model. timateofσv,∗ andthevelocitydispersion ofsource1inferred Inmultiple lens-plane theory,the relationbetweenthe from the Tully-Fisher relation (Moran et al. 2007) are over- angular position ~θ of a light ray in the j-th lens plane laid as a point with error bar. Right panel: contours in and the angular pojsition in the j =1 image plane is: parameterspaceofthesecondsourceredshiftzs2 andtheve- locity dispersion of the first source σSIE,s1. The recovered j−1 D zs2 strongly depends on the mass enclosed in source 1. In θ~ (~θ )=~θ ijα~ˆ(~θ ). (4) both panels confidence levels mark the 68.3, 95.4 and 99.3% j 1 1 i − D enclosed probability. Xi=1 j The last lens plane N can be identified with the source A pixelised source plane inversion for both of these best plane such that ~θ = β~. In Eq. 4, as compared to fitmodelsisshowninFig. 10. Familyimodelsareshown N Bartelmann (2003), we did not consider the reduced de- inthetoprowandfamilyiiinthe bottomrow. Notethe flection which introduces an unnecessary extra D /D very complex systems of caustic and critical lines pro- is s term in the sum. Likewise, the sign convention for the duced by this multiple lens plane system. It is difficult deflection is different than Bartelmann (2003). There- to favor either of these models based on a visual inspec- forefortwodistinctpositionsθ~ andθ~ comingfromtwo tionand either regionon the parameter space has about 1 2 thesamestatisticalweight(fractionofMCMCsamples). distinctsourceplanepositionsβ~ andβ~ respectively,we 1 2 can write: D β~ =~θ ls1 α~ˆ(~θ ) (5) 1 1 1 − D s1 D D D β~ =~θ ls2 α~ˆ(~θ ) s1s2 α~ˆ ~θ ls1 α~ˆ(~θ ) β~ (6). 2 2 2 s1 2 2 1 − D − D (cid:18) − D − (cid:19) s2 s2 s1 In these equations, αˆ is the deflection produced by the main lensing galaxy (lying in the plane that also defines the image plane) and αˆ is the perturbing deflection s1 produced by source 1 (lens 2) onto source 2. Note that parameterslikethecenterofsource1enterthemodeling scheme both as source- and lens-plane parameters. This is clearly visible in the brackets for the argument of αˆ s1 that contains β~ , the position of source 1 in the source 1 plane. The constraints obtained on the equivalent velocity dispersion parameter of the main lens σ and that SIE of source 1 σ are shown in the left panel of Fig. Fig. 10.— Top panels: Best fit family i model image and SIE,s1 9. We clearly see two kinds of solutions: one (fam- sourceplanereconstructions. Fromlefttorightreconstructed ily i) has a high lens velocity dispersion (and slope image plane, residual (data–model), and source 2 plane at γ′ 1.96, nearly isothermal) and little mass in source redshiftzs2 =3.30. Bottom panels: idemforthebestfitfam- 1,w∼hereasthe otherfamily(ii)hasalowermainlensve- ily ii models (with zs2 = 2.75). Note the complex critical locitydispersionandmoremassinsource1. Wemeasure and caustic curves for the zs2 source plane due to the multi- ple lens plane configuration produced by source 1. For both (σ ,σ )=(295+3.5,56 30)kms−1forfamilyiand SIE SIE,s1 −5.0 ± modelsthereconstructionissatisfyingandproducesveryfew (σ ,σ )= (247.3+8.5,104+21)kms−1 for family ii. residuals. SIE SIE,s1 −5.7 −26 10 Gavazzi et al. The left panel of Fig. 9 also shows the aperture- multiple lensingsystems toprobe the cosmology. Future corrected SDSS-inferred velocity dispersion of the lens space-based missions like DUNE or JDEM should pro- σ = 284 24kms−1 which seems to favor family i vide us with tens of thousands of lenses, among which v,∗ solutions, ba±sed on the earlier SLACS results of a gen- several tens would be double source plane systems. We eral agreement between stellar velocity dispersion and also assume that redshifts will be available, from space- σ . In addition, we can get further external infor- or ground-based spectroscopic follow-up. SIE mation on the mass of source 1, by extrapolating the First, we summarize the error budget expected for a Tully-Fisher relation found by Moran et al. (2007) at typicaldoublesourceplanesystem. Asdescribedbefore, z 0.5 for late-type galaxies. In the field, they found themainquantityofinterestistheratioofdistanceratios th∼at at absolute magnitudes of V 19.7, the maxi- parameterη (Dls/Dos)2/(Dls/Dos)1, wheresource2is mum rotation velocity is log(2V ∼) =− 2.2 0.1. As- thefurthesto≡ne. Forsimplicity,weassumethatthemain max suming V √2σ , this translates into a±n estimate lens,the firstsource,andthe secondsourceareperfectly max SIE σ 59 ≃13kms−1. Another piece of information aligned onto the optical axis, resulting in two complete SIE,s1 ≃ ± concentric rings of radius θ and θ . The lens equation comes fromweak lensing results at intermediate redshift 1 2 for each source plane reads: (0.2 < z < 0.4) by Hoekstra et al. (2005), who found that galaxies with magnitude V 5logh 19 have cvoirriraelspmoansdsesstoMlvoirg(≃2V1.50)+−00=..96942.×201−0110.h0−7901,Min⊙≃gow−ohdicahgarelseo- β1=θ1−(Dls/Dos)1αtot(θ1)=0, (7a) max β =θ (D /D ) α (θ )=0. (7b) ± 2 2 ls os 2 tot 2 ment with Moran et al. (2007). These two arguments − alsoseemtofavorfamilyisolutions,i.e. thosewithmore We consider again the general power-lawsurface mass mass in the main lens and less in source 1. distribution of Eq. (1) producing deflections α1 and α2 The right panel of Fig. 9 shows the important degen- onsource1andsource2lightrays. Forsource2wemust eracy between the redshift of source 2 and the velocity addαp the small perturbing deflection11 due to source1 dispersionofsource1. Wecanseethatthemoremassive and experienced by source 2 only. Combining Eq. (7a) source 1, the lower z must be. This demonstrates that and Eq. (7b) gives: s2 any cosmographic test based on multiple source plane lens systems should carefully consider the mass in the θ γ′−1 1 2 η = . (8) froaryesgcroomunindgsforuormcethaesmaossigtndiifisctaannttspoeurrtcuer.bAatdiodninognalsiughbt- (cid:18)θ1(cid:19) 1+ DDsl1ss22ααP2 stantialamountofmassinsource1significantlychanges This equation shows the importance of the perturba- theinferredredshiftofsource2eitherforfamilyimodels tion. Ifoneaimsatconstrainingη withinterestingaccu- which yields zs2 = 2.6+−10..07 or family ii models yielding racy (i.e. error smaller than 0.01), the small perturbing z = 3.8+1.9 . Marginalizing over the whole posterior terminthedenominatorofsecondpartontherighthand s2 −1.5 PDF gives z =3.1+2.0 . side of Eq. (8) should be smaller than 0.01. Keeping s2 −1.0 in mind that for lensing potentials close to isothermal, α σ2, and that the typical velocity dispersion of the TABLE 3 ma∝in lens is about σ 250kms−1, it is important to Best-fit modelparametersforSDSSJ0946+1006usinga control and correct pe≃rturbing potentials with velocity compounddoublelens plane. dispersion as small as σ = σ/10 30kms−1 for values p ∼ Parameter familyi familyii global Ds1s2/Dls2 0.5. ≃ b∞ [arcsec] 2.65+−00..0170 1.91+−00..0076 1.98+−00..6191 Next, differentiating Eq. (8), and writing r ≡ θ2/θ1, γ′ 1.96+0.03 2.23+0.03 2.18+0.07 one can infer the fractional error on η: −0.02 −0.05 −0.22 axisratioq 0.889+0.057 0.816+0.129 0.879+0.067 −0.016 −0.027 −0.083 PA0 −15.9+−91.25.2 −17.9+−91.72.3 −17.0+−91.53.5 δ 2 δ 2 γPeAxtext 0−.02679.+−6+−00..6600..101769 0−.02869.+−5+−00..6600..212762 0−.02872.+−0+−00..6600..212766 (cid:18) ηη(cid:19) = (γ′−1)2(cid:18) rr(cid:19) +(lnr)2δγ2′+ zσsS2IE,s1 [kms−1] 526..66+−−+1032..0707..36 1038.8.9+−−+1111..9859..09 934..10+−−+2124..6600..76 4 2 (cid:18)δσσp(cid:19)2 . (9) σSIE [kms−1] 295+−34 246+−75 254+−4131 (cid:16)1+ DDsl1ss22σσp22(cid:17) p Best fit model parameters and 68.4% confidence limits. An- The first contribution is the relative measurement er- gles are in degrees oriented from North to East. ror on the ratio of Einstein radii, with typical values 0.001 δ /r 0.03 for deep space based imaging. The r ≤ ≤ second term captures our prior uncertainty on the slope 5.3. Future outlook: cosmography with many double ofthedensityprofile(forexampleKoopmansetal.(2006) source plane lenses measured γ′ 2.01andanintrinsicscatterδγ′ 0.12). h i≃ ≃ In 5.1 we exploredthe possibility of constrainingcos- Finally,the thirdtermrepresentsourpriorknowledgeof molog§y with SDSSJ0946+1006,and came to the conclu- the mass of the perturber, which can be based, for ex- sion that the errors are too large for this to be interest- ample, on the Tully-Fisher relation. Moranet al. (2007) ing. In 5.2 we saw that the mass of the closest source must be§taken into account as a perturbation along the 11 We assumethat the non-linear coupling between lens planes can be neglected, i.e. the perturbation of source 1 is small com- double source plane optical bench. Here we attempt to pared to the deflection from the main lens on source 2 light rays: address the possibility of using large numbers of such αp≪α2≃θ2 .