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COSMOLOGICALAPPLICATIONSOFGRAVITATIONALLENSING Y. MELLIER 9 9 Institut d’Astrophysique de Paris 9 and Observatoire de Paris, DEMIRM 1 98 bis Boulevard Arago n 75014 Paris a France J 1 1 1 1. Introduction v 6 Because gravitational lensing provides a unique tool to probe almost di- 1 1 rectly the dark matter, its use for cosmology is of considerable interest. 1 The discovery of giant arc(let)s in clusters of galaxies (Soucail et al 1987, 0 Lynds & Petrosian 1986, Fort et al 1988) or Einstein rings around galaxies 9 9 (Hewitt etal1988), andthespectroscopicproofsthatthey areproducedby / gravitational lensing effects (Soucail et al 1988) have revealed that gravita- h p tional distortion can probe with a remarkable amount of details the mass - distributionofclusters(Tysonetal1990,Kaiser&Squires1993)andgalax- o r ies (Kochanek 1990). t s The applications of gravitational lensing to cosmology are so important a that one cannot ignore them in a course dedicated to observational cos- : v mology. The most obvious applications are the determination of the mass i X of gravitational systems, because the total mass inferred from a simple r gravitational lens model is remarkably robust. In this respect microlens- a ing experiments as well as strong and weak lensing studies provide the most powerful techniques to probe the dark matter of the Universe from the Jupiter-like planets up to large-scale structures. That is more than 16 orders of magnitude in size, more than 19 orders of magnitude in mass, and more than 25 orders of magnitude in density contrast! These limits are only due to technical limitations of present-day instrumentation, and in principle gravitational lensing can probe a much broader range. There- fore, measuring the mass distribution with gravitational lensing can put important constraints on the gravitational history of our Universe and the formation of its structures and its virialized gravitational systems. 2 Y. MELLIER Gravitationallensingalsocanprovidevaluableconstraintsonthecosmolog- ical parameters. For examples, the fraction of quasars with multiple-image, or the magnification bias in lensing-clusters depend on the curvature of the Universe.Deep surveysdevotedtotheselenseshavealreadyputboundaries on the cosmological constant λ. Furthermore, the measurement of time de- lays of transient events observed in multiple images of lensed sources can potentially provide useful constraints on the Hubble constant, H0, on cosmological scales where distortion of the measurements from any local per- turbation is negligible. Of equal importance, gravitational lenses can also be used as gravitational telescopes in order to observe the deep Universe. When strongly magnified, detailed structures of extremely high redshift galaxies can be analyzed spectroscopically in order to understand the dynamical stage and the merging history of the young distant galaxies (Soifer et al 1998) . Furthermore, the joint analysis of the dark matter distribution in clusters and the shapeof the lensed galaxies can be used to recover their redshift distribution. In this course, I will focus on some applications to cosmology. However, in order to avoid self-duplications, I will only address some aspects about themassdetermination fromgravitational lensingstudies.Theother topics more detailed presentations (namely, distant galaxies, cosmological parameters, lensing on the CMB) or more detailed presentations can be found in Bernardeau (this proceedings), Blandford & Narayan (1992), Fort & Mel- lier (1994), Mellier (1998, 1999), Refsdal & Surdej (1994) and obviously in the textbook written by Schneider, Ehlers & Falco (1992). The microlens- ing experiments will be also presented elsewhere (Rich, this proceedings). Since F. Bernardeau has already discussed the theoretical aspects in his lecture, I will only make some addenda on some specific quantities in order to clarify the order of magnitude, but I assume that all the basic concepts are know already. 2. Some important quantities and properties 2.1. IMAGE MULTIPLICITY AND EINSTEIN RADIUS LetusassumethelensbeingapointmassofmassM andaone-dimensional configuration. In that case, the solutions for θ are given by I D 4GM LS θ = θ + (1) S I D D c2θ OL OS I which in general has 2 solutions. This illustrates the fact that gravitational lenses can producemultiple images. In the special case of a point mass, the divergence of the deviation angle at the origin is meaningless. For a more realistic mass density, the divergence vanishes and this produces another COSMOLOGICAL APPLICATIONS OF GRAVITATIONAL LENSING 3 Figure1. Simplelensconfigurationwhichexplainsthebasicnotationsusedinthislecture (see Bernardeau for more details). solution for the images close to the center. This reflects a more general theorem that the number of lensed images is odd for non-singular mass density (Burke 1981). A critical case occurs when θ = 0, that is when S there is a perfect alignment between the source, the lens and the observer. The positions of the images are degenerated and form a strongly magnified circle at the Einstein radius θ : E 4GM D 1/2 LS θ = . (2) E c2 D D (cid:20) OL OS(cid:21) For example, for a star of 1 M⊙ at distance D=1 kpc, θE = 0.001 arcsec, − for a galaxy of 1012 M⊙ at D=1 Gpc, θE = 1 arcsec, − andforaclustersof1014 M⊙ atz = 0.3andsourcesatzS = 1,θE = 30 − arcsec. When the source if slightly off-centered with respect to the axis defines by the lens and the observed, a circular lens produces two elongated arcs diametrally opposite with respect to the center of the lens. 2.2. CRITICAL MASS By definition, the magnification matrix is given by A = dθ /dθ . This S I defines the convergence, κ, and the shear,(γ1,γ2), 1−κ−γ1 −γ2 = 1−∂xxϕ −∂xyϕ , (3) γ2 1 κ+γ1 ∂xyϕ 1 ∂yy (cid:18) − − (cid:19) (cid:18) − − (cid:19) where “∂ ” denotes the partial derivative of the projected potential with ij respect to the coordinates ij. The magnification is given by 1 1 µ = = , (4) detA (1 κ2) γ2 | | | − − | where the shear amplitude γ = γ2+γ2 expresses the amplitude of the 1 2 anisotropic magnification. On thqe other hand, κ expresses the isotropic part of the magnification. In is worth noting that 2κ = ∆ϕ = Σ/Σ , crit is directly related to the projected mass density. It may happen that the determinantvanishes.Itthatcase,themagnificationbecomesinfinite.From an observational point of view, these cases correspond to the formation of very extended images, like Einstein rings or giant arcs. The points of the 4 Y. MELLIER image plane where the magnification is infinite are called critical lines. To these critical lines correspond points in the source plane which are called caustic lines. The critical density is c2 D OS Σ = (5) crit 4πGD D LS OL expresses the capability of a gravitational system to producestrong lensing effect (Σcrit 1). If we scale to 2H0/c2 in order to define the “normalized” ≥ angular distances d , then ij H0 dos −2 Σ 0.1 g.cm (6) crit ≈ 50km/sec/Mpc d d (cid:18) (cid:19) ls ol For example, for a lensing-cluster at redshift z = 0.3 and lensed sources L at redshift z = 1, d /(d d ) 3. If the cluster can be modeled by a S os ls ol isothermal sphere with a core rad≈ius Rc and with M(Rc) = 2 1014 M⊙, × then For R =250 kpc, Σ =0.05 g.cm−2, c crit − For R =50 kpc, Σ =1. g.cm−2 . c crit − Hence, the existence of giant arcs in clusters implies that clusters should be strongly concentrated. 2.3. RELATION WITH OBSERVABLE QUANTITIES Since, to first approximation, faint galaxies look like ellipses, their shapes can be expressed as function of their weighted second moments, S(θ)(θ θC)(θ θC)d2θ M = i− i j − j , (7) ij S(θ)d2θ R where the subscripts i j denote thRe coordinates θ in the source and the image planes, S(θ) is the surface brightness of the source and θC is the center of the source. Since gravitational lensing effect does not change the surface brightness of the source (Etherington 1933), then, if one assumes that the magnification matrix is constant across the image, the relation between the shape of the source, MS and the lensed image, MI is MI = A−1 MS A−1 (8) Therefore, the gravitational lensing effect transforms a circular source into an ellipse. Its axis ratio is given by the ratio of the two eigenvalues of the COSMOLOGICAL APPLICATIONS OF GRAVITATIONAL LENSING 5 magnification matrix. The shape of the lensed galaxies can then provide informationaboutthesequantities.However, thoughγ1 andγ2 describethe anisotropic distortion of the magnification, they are not directly related to observables (except in the weak shear regime). The reduced complex shear, g, and the complex polarization (or distortion), δ, γ 2g 2γ(1 κ) g = ; δ = = − , (9) (1 κ) 1+ g 2 (1 κ)2+ γ 2 − | | − | | are more relevant quantities because δ can be expressed in terms of the observed major and minor axes aI and bI of the image, I, of a circular source S: a2 b2 − = δ (10) a2+b2 | | In this case, the 2 components of the complex polarization write: M11 M22 2M12 δ1 = − ; δ2 = , (11) Tr(M) Tr(M) where Tr(M) is the trace of the magnification matrix. For non-circular sources,itispossibletorelatetheellipticity oftheimageǫI totheellipticity ofthelensedsource,ǫS.Inthegeneralcase,itdependsonthesignofDet(A) (that is the position of the source with respect to the caustic lines) which expresses whether images are radially or tangentially elongated: 1+bI/aI ǫS +g ǫI = e2iϑ = for Det(A) > 0 , (12) 1+bI/aI 1 g∗ǫS − and 1+ǫS∗g ǫI = for Det(A) < 0 . (13) ǫS∗+g∗ 3. Academic examples 3.1. THE SINGULAR ISOTHERMAL SPHERE The projected potential at radius r, ϕ(r), of a singular isothermal sphere with 3-dimension velocity dispersion σ is σ2 D LS ϕ = 4π r (14) c2 D OS and the images are described by the lensing equation σ2 D θ LS I θ = θ 4π (15) S I − c2 D θ OS I | | 6 Y. MELLIER Figure 2. Gravitational distortion produced by an elliptical potential as a function of source position. The top left panel shows the shape of the source inthe source plane. The secondpanelshows10positionsofthesourceinitssourceplane(referencedfrom1to10) withrespect to asimulatedcluster lens. Thethinlinesshowthe innerandouter caustics. Panels 3 to 12 show the inner and outer critical lines and the shape of the image(s) of the lensed source. Positions 6 and 7 correspond to cusp configurations, and position 9 is typical fold configuration. On the fifth panel we see two inner merging images forming a typical radial arc (from Kneib 1993, PhD thesis). and the magnification matrix 1 0 σ2 D 1 (16) LS  0 1 4π  − c2 D θ OS I | |   Hence, there is only one critical line which is given by the Einstein radius σ2 D σ 2 LS θ = 4π 16” (17) SIS c2 D ≈ 1000km.sec−1 OS (cid:18) (cid:19) for an Einstein de Sitter universe, with z = 0.3 and z = 1. The total L S mass included within the radius θ is then 2 θ σ 13 −1 M(θ)= 5.7×10 M⊙h50 16” 1000km.sec−1 (18) (cid:18) (cid:19)(cid:18) (cid:19) From Eq.(28) and (29), it is obvious that the magnification writes θ I µ(θ )= (19) I θ θ I SIS − Thesingular isothermal spherepermits to keep in mind thevarious properties of gravitational lenses and order of magnitude estimates of the associated physical quantities. However, more complex lenses can produce somewhat different configurations of strong lensing. Some example are shown in Figure 2, where a series of arcs generated by an elliptical potential well is displayed. We see that various kind of multiple images can be produced, with some strange radial arcs for some configurations. Each panel can be foundintheuniversesothiskindoftemplate of lensfeatures can behelpful for theunderstandingof the lens modeling. This has been extensively used, in particular on HST images (see Fort & Mellier 1994, Kneib et al 1996, Natarajan & Kneib 1997, Mellier 1998 and references therein). 3.2. THE GENERAL CASE OF AN AXIALLY SYMMETRIC LENS Assuming the rescaled angle writes (see Bernardeau, this proceedings) α= ∇ϕ , ∆ϕ= 2κ(x) , (20) COSMOLOGICAL APPLICATIONS OF GRAVITATIONAL LENSING 7 for an axially symmetric potential, we have dϕ 1 d dϕ α = , x = 2κ(x) , (21) dx xdx dx (cid:18) (cid:19) and the mass at radius x is m(x) = m(x ) = m(x). Hence | | dm dϕ 2 x ′ ′ ′ = 2xκ(x) = = κ x xdx . (22) dx ⇒ dx x 0 Z (cid:0) (cid:1) Therefore, in general the magnification matrix writes A= I − mx(4x) x222−x1xx212 −x212x1xx222 − x13 dmdx(x) xxxx212 xxxx222 , (cid:18) − − (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (23) where is the identity matrix. The expression of the shear terms is then I immediate, and the total magnification wirtes: m d m(x) Det(A) = 1 1 (24) − x2 − dx x (cid:18) (cid:19)(cid:18) (cid:18) (cid:19)(cid:19) where the first term on the right-hand part provides the position of the tangential arcs, whereas the second term provides the location of the radial arcs. These equations are useful in order to compute the lensing properties of mass distributions, like the universal profile discussed in Section 4.1.2 . 4. Astrophysical examples 4.1. MEASURING THE MASSES OF CLUSTERS OF GALAXIES 4.1.1. The case of MS2137-23 Strong lensing refers to lensing configurations where the source is located closetoacausticlineandproducesarcsorrings.Inadditiontotheaccurate estimate of the total mass, strong lensing features can probe some details on the mass density profile. A nice example is the case of the lensing cluster MS2137-23 which contains a radial and a tangential arc. A simple investigation already permits to have some constraints on the lens configuration. First, since there is no counter-arc associated to the tangential arc on the diametrally opposite side of the cluster center, it is unlikely that the projected mass density is circularly distributed. Second, since the tangential arc and the radial arc are on the same side, it is unlikely that they are images of the same source.Aninterestingtentativetomodelthelensistoassumethatthedark matter follows the light distribution of the central galaxy (see Figure 3). The isophotes then provide the center, the orientation and the ellipticity Figure3. ModelofMS2137-23.Thisclusterisatredshift0.33andshowsatangentialarc andthefirstradialarceverdetected. Uptonow itisthemostconstrained clusterandthe fi8rst one where counter images were pYr.edMicEteLdLbIeEfoRre being observed (Mellier et al 1993). The external and the dark internal solid lines are the critical lines. The internal grey ellipseandthe diamondare the caustic lines.Thethin isocontours shows the positions of the arcs and their counter images. of light as constraints to dark matter. Let us assume that the projected potential as the following geometry 2 r ϕ(r,θ) = ϕ0 1+ (1 ǫcos(2θ)) (25) s (cid:18)rc(cid:19) − where rc is the core radius, ǫ the ellipticity, and ϕ0 the depth of the potential, respectively. For an elliptical lens, the magnification matrix can be written 1∂ϕ 1 ∂2ϕ 1 0  − r ∂r − r2 ∂θ2 ∂2ϕ  (26) 0 1  − ∂r2      which gives immediately the position or the radial and tangential arcs, ∂2ϕ 1∂ϕ 1 ∂2ϕ 1 = 0 ; 1 = 0 (27) " − ∂r2# " − r ∂r − r2 ∂θ2# R T From these two constraints, we can have a useful information of the core radius itself (Mellier et al 1993): 2 3/2 r D D r R +1 = OLR OST T +1 (28) (cid:18)rc (cid:19) (cid:18)DOSR(cid:19)(cid:18)DOLT (cid:19)"(cid:18)rc(cid:19) # where the subscripts R and T denote the radial and tangential arc respectively. Hence, provided the redshifts of the lens and the two sources are known, one can infer the core radius,or more generally the steepness of the dark matter density. The modeling of the central regions of clusters of galaxies shows that the geometry of the dark matter follows the geometry of the diffuse light associated to the dominant galaxies. This is a remarkable success of the strong lensing analysis. However, there are still some uncertainties about the total mass which does not agree with the mass inferred from X-ray analysis. The mass-to-light ratio ranges in the same domain ( 100-300), but one can ≈ find frequently a factor of two difference. 4.1.2. The X-ray/lensing mass discrepancy As it is illustrated in Figure 4, it turns out that the peaks of dark matter revealed by giant arcs in lensing-clusters correspond to the the peaks if X-ray emission as well as the location of the brightest components of COSMOLOGICAL APPLICATIONS OF GRAVITATIONAL LENSING 9 TABLE1. Resultsobtainedfromstronglensinganalysesofclusters.These are only few examples. σobs is the velocity dispersion obtained from spec- troscopy of cluster galaxies. σDM is the velocity dispersion of the dark matter obtained from themodeling of the lens. Cluster z σobs σDM M/L Scale (kms−1) (kms−1) (h ) (h−1 Mpc) 100 100 MS2137-23 0.33 - 900-1200 680-280 0.5 ) A2218 0.18 1370 1000 200 0.9 ≈ A2390 0.23 1090 1260 250 0.4 Cl0024+17 0.39 1250 1300 >200 0.5 A370 0.38 1370 850 >150 0.5 Cl0500-24 0.316 - <1200 <600 0.5 optical images. On the other hand, Miralda-Escudé & Babul (1995) have pointed out an apparent contradiction between the mass estimated from X-ray data and the lensing mass (M 2 3M ). Further works lensing X ≈ − done by many groups lead to somewhat inconclusive statements about this contradiction. Böhringer et al (1998) find an excellent agreement between X-ray and lensing masses in A2390 which confirms the view claimed by Pierre et al (1996); Gioia et al (1998) show that the disagreement reaches a factor of 2 at least in MS0440+0204; Schindler et al (1997) find a factor of 2-3 discrepancy for the massive cluster RXJ1347.4-1145, but Sahu et al (1998) claim thatthedisagreement ismarginalandmay notexist;Otaetal (1998) and Wu & Fang (1997) agree that there are important discrepancies in A370, Cl0500-24 and Cl2244-02. There is still no definitive interpretation of these contradictory results. It could be that the modeling of the gravitational mass from the X-ray distribution is not as simple. By comparing the geometry of the X-ray isophotes of A2218 to the mass isodensity contours of the reconstruction, Kneib et al (1995) found significant discrepancies in the innermost parts. The nu- merous substructures visible in the X-ray image have orientations which do not follow the projected mass density. They interpret these features as shocks produced by the in-falling X-ray gas, which implies that the current description of the dynamical stage of the inner X-ray gas is oversimplified. Recent ASCA observations of three lensing-clusters corroborate the view 10 Y. MELLIER Figure4. AnalysisofthethematterdistributionintheclusterA370usingstronglensing features (arcs and arclets). The top right panel shows the R-light distribution from the isoluminositycontours ofgalaxies. Thetopleftpanel showsthe numberdensity contours. The bottom right is a B CCD image of the clusters with the X-ray luminosity contours overlayed.Thearcletsdirectlyshowtheshapeoftheprojectedmassdensity.Inparticular, we see that the arclet pattern indicates the presence of an extension toward the eastern region, which is also seen in the isoluminosity contours in the R-band and the X-ray maps. This clearly shows that in the center of this cluster light from galaxies and from the hot gas trace the mass. Note the arc reconstruction and the mass model in the fourth panel (the triplet B2-B3-B4 is discussed by Kneib et al. 1993). that substructures are the major source of uncertainties. In order to study this possibility in more details, Smail et al (1997) and Allen (1998) have performed a detailed comparison between the lensing mass and X-ray mass for a significant number of lensing clusters. Both works conclude that the substructures have a significant impact on the estimate of X-ray mass. More remarkably, the X-ray clusters where cooling flows are present do not show a significant discrepancy with X-ray mass, whereas the others X-ray clusters do (Allen 1998). This confirms that the discrepancy is certainly due to wrong assumptions on the physical state of the gas. The interpretation of this dichotomy in cluster samples may be the following. Clusters with cooling flows are compact and rich systems which probably have probably virialised and have a well-defined relaxed core. Therefore, when removing the cooling flow contribution, the assumptions thatthe gas is in hydrostatic equilibriumis fullysatisfied. Conversely, non-cooling flow clusters are generally poor, do have lot of substructures and no very dense core dominates the cluster yet. For these systems, the gas cannot be described simply (simple geometry, hydrostatic equilibrium) and the oversimplification of its dynamical stage produces a wrong mass estimator. This interpretation needs further confirmations. However, from these two studies we now have the feeling that we are now close to understand the origin of the X-ray and lensing discrepancy. An alternative has been suggested by Navarro, Frenk & White (1997) who proposed that the analytical models currently used for modeling mass distributions may be inappropriate (hereafter NFW). They argue that the universal profile of the mass distribution produced in numerical simulations of hierarchical clustering may reconcile the lensing and X-ray masses. This kind of profile must beconsidered seriously because the universal profile is a natural outcome from the simulations which does not use external prescriptions. For the reader who want to look more deeply at the lensing properties of this profile, it is interesting to describe the NFW properties into more details. Let us assume that the cluster 3-dimension mass density has the