The lensing information provided by multiple images, arclets, and statistical
distortions can all be formulated as linear constraints on the arrival-time
surface, and hence on the mass distribution. This reduces cluster lens
reconstruction from combined strong and weak lensing to a standard type of
inversion problem. Moreover, the mass sheet degeneracy is broken if there are
sources at different redshifts.Comment: 8 pages incl 6 figures. Invited talk at "Gravitational Lensing:
  Recent Progress and Future Goals", Boston University, MA, July 1999, eds.
  T.G. Brainerd and C.S. Kochane

AbdelSalam, Hanadi

Saha, Prasenjit

Williams, Liliya L. R.

English

arXiv

Abstract: The lensing information provided by multiple images, arclets, and statistical distortions can all be formulated as linear constraints on the arrival-time surface, and hence on the mass distribution. This reduces cluster lens reconstruction from combined strong and weak lensing to a standard type of inversion problem. Moreover, the mass sheet degeneracy is broken if there are sources at different redshifts

Saha, P.

Williams, L. L. R.

Abdel-Salam, H. M.

University of Groningen

   University of GroningenCluster Reconstruction from Combined Strong and Weak LensingSaha, P.; Williams, L. L. R.; Abdel-Salam, H. M.Published in:ArXivDOI:10.1.1.35.1213IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Saha, P., Williams, L. L. R., & Abdel-Salam, H. M. (1999). Cluster Reconstruction from Combined Strongand Weak Lensing. Manuscript submitted for publication. https://doi.org/10.1.1.35.1213CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Cluster Reconstruction from Combined Strong and WeakLensingPrasenjit SahaDepartment of Physics, University of Oxford, UKLiliya L.R. WilliamsDepartment of Physics and Astronomy, Univ. of Victoria, BC, CanadaHanadi M. AbdelSalamKapteyn Sterrewacht, Rijksuniversitaet Groningen, The NetherlandsAbstract. The lensing information provided by multiple images, ar-clets, and statistical distortions can all be formulated as linear constraintson the arrival-time surface, and hence on the mass distribution. This re-duces cluster lens reconstruction from combined strong and weak lensingto a standard type of inversion problem. Moreover, the mass sheet de-generacy is broken if there are sources at different redshifts.This paper presents a reconstruction technique based on these ideas,and a mass map and discussion of Cl 1358+62.1. IntroductionThe different regimes of cluster lensing—the inner region where multiple images,including giant arcs are found, further out where highly sheared but singly im-aged arclets are found, and the outer regions showing statistical shear—have inthe past been approached with quite different modeling methods. But these ap-parently separate regimes can be studied in a unified way. The key is to expresslensing information in terms of the arrival surface.In earlier work (see AbdelSalam et al. 1998) we combined the multiple-image and arclet regimes and argued that extending to statistical shear was astraightforward algorithmic issue. This paper makes that extension.2. The arrival-time surfaceThe creature we will mostly be concerned with is the scaled arrival-time surface(θ), and it is expressed as follows:(θ) = 12(θ− β)2 −(Dls=Ds)∫ln jθ− θ0j(θ0) d2θ0 (1)= 12(θ− β)2 − 2(Dls=Ds) r−2(θ): (2)1θy’θx’                                                                                                  ω θθxyFigure 1. Schematic of an arclet elongated with position angle !,with (x′ ; y′) being a coordinate system aligned with the elongation.As usual, θ and β are angular locations on the image and source planes respec-tively, and the D’s are angular diameter distances in units of c=H0. However, is the convergence for sources at infinity, (hence the factors of Dls=Ds) to makedealing with multiple source redshifts more convenient. The form of Eq. (1) isthen familiar, while Eq. (2) is just a shorthand for the same thing with r−2denoting an ‘inverse Laplacian operator’ in two dimensions.To get back to physical arrival time and surface density we usetime = h−1z˜  80 days arcsec−2  (θ) (3)surface density = h−1z˜  1:2 1011M arcsec−2  (θ) (4)where z˜ = (1 + zl)Dl but is ’ zl.Lensing data constrain the arrival-time surface and hence the mass distri-bution (θ) in various ways. We can identify three types of constraints: on thevalues of the arrival time surface at particular points, on the first derivative, andon the second derivative.The first type is when we know the height difference between two or morepoints on the arrival-time surface. This is all-important in time-delay quasarsand H0 measurements, because time delays supply just this height-differenceinformation. But it is not relevant to cluster lensing, at least not yet, thoughwe can always hope for a supernova in a giant arc.The second type of constraint comes from stating Fermat’s principle at eachimage location, i.e.,r(θimage) = 0: (5)Since image positions θimage can generally be measured very accurately, theEq. (5) tells us that the arrival-time surface has zero gradient at some knownpoint. This type of constraint is only useful if we have multiple images; in thatcase we have 2himagesi equation of the form (5) but only two unknown sourcecoordinates to solve for, giving us some net constraints on (θ). In general thereare 2(himagesi − hsourcesi) constraints.The third type of constraint involves the curvature or second derivative ofthe arrival-time surface, and may come from either arclets or from statisticaldistortions.Consider first a situation like Fig. 1, where an arclet is observed elongatedwith position angle !. If we are confident that the arclet has been elongated bya factor of at least k, we can consider the coordinate system (x′ ; y′) aligned2with the elongation and writek∣∣∣∣∣ @2@2x′ (θimage)∣∣∣∣∣ ∣∣∣∣∣ @2@2y′ (θimage)∣∣∣∣∣ : (6)Further, since ! is known, we can transform Eq. (6) to the unrotated (x; y)coordinates. It is not necessary to have highly accurate values of the elongationor its orientation !—all we have to do is set k conservatively enough that theinequality (6) is valid. The arclet used may itself be part of a multiple-imagesystem; if so we will have to guess the parity to remove the absolute value signsin (6).The same idea can be applied to statistical distortion. In this case the dataprovide an estimate and uncertaintyγi1−  = gi ∆gi; (7)where γi and  are components of the second derivative of (θ). We can rewritethis in various ways, such as(1− (θimage))gi −∆gi  γ(θimage)  (1− (θimage))gi + ∆gi: (8)(We have assumed ∆gi is negligible here.)A key feature of the constraint equations (5), (6), and (8) is that they areall linear in the unknowns β and (θ). Which is to say1( )=( ) 0 (9)and moreover this is true for all lensing regimes: multiple-image, arclet, andstatistical shear.3. Mass reconstructionEq. (9) summarizes the advantage of casting the observational information asconstraints on the arrival-time surface. Lens reconstruction is reduced to a linearinversion problem. It is (as symbolized by the 2  5 matrix) a highly underde-termined problem, so in order to reconstruct the lens it is necessary to add extrainformation (sometimes called a prior). Also as indicated in the 25 matrix, thedata enter into the linear operator—an unusual complication. However, as we1With one exception: scalar magnification data are quadratic in κ and so (9) does not apply.Taylor and collaborators (see e.g., Dye & Taylor 1998) have developed reconstruction techniquesfor this case. However, if magnification information is present along with shear information,together they give (at least in principle) the tensor magnification which is linear in κ, and so(9) again applies.3Figure 2. At fixed redshift, the mass sheet degeneracy leaves 1 − uncertain by a multiplicative factor, while the monopole part of  isundetermined outside the region of the images. As a result, the twomass profiles shown will produce identical multiple-image and sheardata out to the cliff.discussed above, the data enter there either as image positions which are knownvery accurately, or as inequalities which can be set conservatively; so this issuedoes not introduce new difficulties.At least four different possibilities for lens reconstruction now suggest them-selves Put (θ) on a grid and regularize. This may be the simplest approach. Use basis function expansions for (θ) and (θ). A Fourier-Bessel expan-sion(θ) =∑mncmnJm(kmn) exp(im)is particularly attractive as it would trivially relate  and . Pixellate (θ) and regularize. This is the one we have implemented. Themass is distributed on square tiles, each having constant but adjustabledensity, and (θ) comes from computing Eq. (1) exactly. Note that al-though the mass distribution is discontinuous, the arrival-time surface issmooth. Unlike in the two previous possible approaches, pixellating themass makes it is easy to enforce non-negativity of the mass distribution. Pixellate (θ) and use maximum entropy. See Bridle et al. (this volume)for an example of this in a somewhat different context.The regularization we applied is to minimize∫ (− hihLiL)2d2θ + 4∫(r2)2 d2θ (10)while of course enforcing all the lensing constraints. The first term in (10) tendsto minimize mass-to-light variation, since hi=hLi is a mean M=L. The secondterm tends to smooth, with  a sort of smoothing scale. By regularizing withrespect to different light distributions we can get an estimate of the uncertainty.Since the regularizing functional (10) is quadratic in , minimizing it sub-ject to the linear lensing constraints is readily implemented through quadratic4Figure 3. Left: Cluster luminosity contours in Cl 1358+62. fromthe catalog of galaxy Vz’s in van Dokkum et al (1998). Likened atthe conference to a left facing seahorse with a ponytail. Right: Shearmap (exaggerated by a factor of 5) from Hoekstra et al (1998). Wehave excised the central region to avoid averaging the shear where itis varying rapidly. Also note that unlike Fig. 14 in Hoekstra et al., wehave binned without smoothing here.programming algorithms. The disadvantage is that the storage needed is ’2hnumber of pixelsi2 limiting us to of order 5000 pixels. The solution is to haveadaptive pixellation (much like tree codes in N -body simulations), with smallerpixels for the inner parts of the cluster and large pixels outside.4. The mass-sheet degeneracyAfter enthusing about how easy lens reconstruction is, it is well to make acautionary remark about the main source of uncertainty. Here again, the arrival-time surface is very useful.If we multiply (θ) by a constant factor, we just stretch the surface verti-cally, and geometrically it is clear that neither image locations nor their relativemagnifications change. More formally, we can rearrange Eq. (2) and drop theirrelevant term 12β2 to get(θ) = 2r−2(1− DlsDs)− 12θ  β: (11)If we now multiply both (1−Dls=Ds) and β by some constant a, the imagestructure will be unchanged; only time delays and total magnification will getmultiplied by a and a−1 respectively (the latter because rescaling β rescalesall the sources). Therefore lens reconstruction from image structure (withoutabsolute magnifications) leaves (1−Dls=Ds) uncertain by a constant factor.This is the mass-sheet degeneracy.5Figure 4. Pixellation used for the mass reconstruction of the Sea-horse cluster. Note that is completely adaptive, and allows us toincrease the resolution at any desired place—in this case the regionaround the red arc. Left: The full field. Right: The inner region,with dots marking features in the red arc.It may appear at first that the mass-sheet degeneracy can be eliminated byrequiring  to vanish at large distances. But outside the observed region, themonopole part of  is completely unconstrained. So the mass-sheet degeneracyis equally effective with a mass disk larger than the observed region. Figure 2illustrates.5. Reconstruction of Cl 1358+62 (The Seahorse Cluster)This cluster seemed the obvious test case for combined strong and weak lensingreconstruction in a relatively large field. The cluster itself is at z = 0:33 and itsinner part lenses a z = 4:92 galaxy into a red arc; this was identified by Franxet al. (1997) who also presented a strong lens model. Hoekstra et al. (1998)measured the shear field in an HST WFPC2 mosaic and reconstructed a massmap from weak lensing. The present work combines both regimes.Figure 3 illustrates the data on this cluster that we have used: a shear fieldderived by binning the individual polarization measurements kindly provided byHoekstra and collaborators and assuming a constant z = 1 for the backgroundgalaxies, and a smoothed Vz light distribution. Figure 4 shows the pixellationwe used.Figure 5 is our mass map, computed by regularizing with respect to thelight and a smoothing scale  changing from ’ 500 in the center to ’ 10 at theedge. Our estimated uncertainty is derived from an ensemble of reconstructionswhere we rotated the light map by arbitrary angles and shifted it randomly byup to 10000 and regularized with respect to these altered light maps. The massmap resembles Fig. 15 of Hoekstra et al. (1998) but tends to be smoother. Alsothe overall normalization is somewhat higher; this is probably due to different6Figure 5. Mass map of the Seahorse cluster. Left: Contours of  insteps of 0.1. Right: Contours of ∆ in steps of 0.025.treatments of the boundary, though with the mass-sheet degeneracy. (The redarc at z = 4:92 compared to z = 1 assumed for the weakly lensed galaxies,reduces the effect of the mass-sheet degeneracy, but does not eliminate it.) Ourcentral density is higher, but this is as expected since inclusion of a multiply-imaged system immediately forces  > 1. There is some indication that themass peak is offset (by some 10s of kpc) to the south of the light peak, a naturalthing to expect if the cluster is asymmetric and galaxy formation is biased. Butthis offset is tentative; in Abell 2218, where the inner region is much richer inlensing and better constrained, the evidence for an offset is more compelling(AbdelSalam et al. 1998).Figure 6 shows the enclosed mass out to different radii and the reconstructedshear. It is surprising that the enclosed mass after angular averaging, looks so‘isothermal’ even though the mass map is very asymmetric. We obtain M =(10  1)  1014M Mpc−1, corresponding to a formal Einstein radius of 4300and a formal isothermal los dispersion of (990  70) km sec−1. The estimatedmass-to-light is (380  60)hML−1V . The CNOC survey (Carlberg et al. 1997)measured 910 km sec−1 for the los velocity dispersion and estimated M=L =229h−1ML−1V .6. DiscussionWe have developed a mass reconstruction technique that deals with all the clus-ter lensing regimes simultaneously, from multiple images in the central regionsto weak shear in the outer regions, and moreover with adaptive resolution. Codeimplementing this is available from the authors. Variants of our technique (sug-gested in Section 3) may also be of interest in future work.At this stage it appears that cluster mass reconstruction is very good atrecovering features—a good example are offsets between mass and light peaks7Figure 6. Left: Shear map. The superimposed ellipses show the ob-served and reconstructed (dotted) shear, both exaggerated by a factorof five. As we might expect, the reconstructed shear appears more cor-related than the observed. The ellipses in the inner region show thereconstructed shear (not exaggerated) at the locations of the red arc.Right: Enclosed mass and estimated uncertainty.in clusters, which may be indirect evidence for biased galaxy formation. Butthe calibration of the mass (even with perfect shear data) is still problematic—the mass sheet degeneracy is a vicious effect. We suggest that recovering theredshift distribution and the absolute magnification of the background galaxies(even with large uncertainties) are the best hope of breaking this degeneracy,and is likely to be very rewarding in future work.Acknowledgments. We are grateful to Henk Hoekstra and collaboratorsfor supplying us with their shear data on Cl 1358+62.ReferencesAbdelSalam, H.M., Saha, P., & Williams, L.L.R. 1998, AJ, 116, 1541Bridle, S.L., Hobson, M.P., Lasenby, A.N., Saunders, R., this volumeCarlberg, R.G., Yee, H.K.C., Ellingson, E. 1997, ApJ, 478, 462van Dokkum, P.G., Franx, M., Kelson, D.D., Illingworth, G.D., Fisher, D.,Fabricant, D. 1998, ApJ, 500, 714Dye, S., & Taylor, A. 1998, MNRAS, 300, L23Franx, M., Illingworth, G.D., Kelson, D.D., van Dokkum, P.G., Tran, K.-V.1997, ApJ, 486, L75Hoekstra, H., Franx, M. Kuijken, K., Squires, G. 1998 ApJ, 504, 6368

Cluster Reconstruction from Combined Strong and Weak Lensing

NARCIS 

   University of GroningenCluster Reconstruction from Combined Strong and Weak LensingSaha, P.; Williams, L. L. R.; Abdel-Salam, H. M.Published in:ArXivDOI:10.1.1.35.1213IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Saha, P., Williams, L. L. R., & Abdel-Salam, H. M. (1999). Cluster Reconstruction from Combined Strongand Weak Lensing. Manuscript submitted for publication. https://doi.org/10.1.1.35.1213CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 27-04-2023Cluster Reconstruction from Combined Strong and WeakLensingPrasenjit SahaDepartment of Physics, University of Oxford, UKLiliya L.R. WilliamsDepartment of Physics and Astronomy, Univ. of Victoria, BC, CanadaHanadi M. AbdelSalamKapteyn Sterrewacht, Rijksuniversitaet Groningen, The NetherlandsAbstract. The lensing information provided by multiple images, ar-clets, and statistical distortions can all be formulated as linear constraintson the arrival-time surface, and hence on the mass distribution. This re-duces cluster lens reconstruction from combined strong and weak lensingto a standard type of inversion problem. Moreover, the mass sheet de-generacy is broken if there are sources at different redshifts.This paper presents a reconstruction technique based on these ideas,and a mass map and discussion of Cl 1358+62.1. IntroductionThe different regimes of cluster lensing—the inner region where multiple images,including giant arcs are found, further out where highly sheared but singly im-aged arclets are found, and the outer regions showing statistical shear—have inthe past been approached with quite different modeling methods. But these ap-parently separate regimes can be studied in a unified way. The key is to expresslensing information in terms of the arrival surface.In earlier work (see AbdelSalam et al. 1998) we combined the multiple-image and arclet regimes and argued that extending to statistical shear was astraightforward algorithmic issue. This paper makes that extension.2. The arrival-time surfaceThe creature we will mostly be concerned with is the scaled arrival-time surfaceτ(θ), and it is expressed as follows:τ(θ) = 12(θ− β)2 −(Dls/Ds)π∫ln |θ− θ′|κ(θ′) d2θ′ (1)= 12(θ− β)2 − 2(Dls/Ds) ∇−2κ(θ). (2)1θy’θx’                                                                                                  ω θθxyFigure 1. Schematic of an arclet elongated with position angle ω,with (θx′ , θy′) being a coordinate system aligned with the elongation.As usual, θ and β are angular locations on the image and source planes respec-tively, and the D’s are angular diameter distances in units of c/H0. However, κis the convergence for sources at infinity, (hence the factors of Dls/Ds) to makedealing with multiple source redshifts more convenient. The form of Eq. (1) isthen familiar, while Eq. (2) is just a shorthand for the same thing with ∇−2denoting an ‘inverse Laplacian operator’ in two dimensions.To get back to physical arrival time and surface density we usetime = h−1z̃ × 80 days arcsec−2 × τ(θ) (3)surface density = h−1z̃ × 1.2× 1011M arcsec−2 × κ(θ) (4)where z̃ = (1 + zl)Dl but is ' zl.Lensing data constrain the arrival-time surface and hence the mass distri-bution κ(θ) in various ways. We can identify three types of constraints: on thevalues of the arrival time surface at particular points, on the first derivative, andon the second derivative.The first type is when we know the height difference between two or morepoints on the arrival-time surface. This is all-important in time-delay quasarsand H0 measurements, because time delays supply just this height-differenceinformation. But it is not relevant to cluster lensing, at least not yet, thoughwe can always hope for a supernova in a giant arc.The second type of constraint comes from stating Fermat’s principle at eachimage location, i.e.,∇τ(θimage) = 0. (5)Since image positions θimage can generally be measured very accurately, theEq. (5) tells us that the arrival-time surface has zero gradient at some knownpoint. This type of constraint is only useful if we have multiple images; in thatcase we have 2〈images〉 equation of the form (5) but only two unknown sourcecoordinates to solve for, giving us some net constraints on κ(θ). In general thereare 2(〈images〉 − 〈sources〉) constraints.The third type of constraint involves the curvature or second derivative ofthe arrival-time surface, and may come from either arclets or from statisticaldistortions.Consider first a situation like Fig. 1, where an arclet is observed elongatedwith position angle ω. If we are confident that the arclet has been elongated bya factor of at least k, we can consider the coordinate system (θx′ , θy′) aligned2with the elongation and writek∣∣∣∣∣ ∂2∂θ2x′ τ(θimage)∣∣∣∣∣ ≤∣∣∣∣∣ ∂2∂θ2y′ τ(θimage)∣∣∣∣∣ . (6)Further, since ω is known, we can transform Eq. (6) to the unrotated (θx, θy)coordinates. It is not necessary to have highly accurate values of the elongationor its orientation ω—all we have to do is set k conservatively enough that theinequality (6) is valid. The arclet used may itself be part of a multiple-imagesystem; if so we will have to guess the parity to remove the absolute value signsin (6).The same idea can be applied to statistical distortion. In this case the dataprovide an estimate and uncertaintyγi1− κ = gi ±∆gi, (7)where γi and κ are components of the second derivative of τ(θ). We can rewritethis in various ways, such as(1− κ(θimage))gi −∆gi ≤ γ(θimage) ≤ (1− κ(θimage))gi + ∆gi. (8)(We have assumed κ∆gi is negligible here.)A key feature of the constraint equations (5), (6), and (8) is that they areall linear in the unknowns β and κ(θ). Which is to say1( )=( ) ′ (9)and moreover this is true for all lensing regimes: multiple-image, arclet, andstatistical shear.3. Mass reconstructionEq. (9) summarizes the advantage of casting the observational information asconstraints on the arrival-time surface. Lens reconstruction is reduced to a linearinversion problem. It is (as symbolized by the 2 × 5 matrix) a highly underde-termined problem, so in order to reconstruct the lens it is necessary to add extrainformation (sometimes called a prior). Also as indicated in the 2×5 matrix, thedata enter into the linear operator—an unusual complication. However, as we1With one exception: scalar magnification data are quadratic in κ and so (9) does not apply.Taylor and collaborators (see e.g., Dye & Taylor 1998) have developed reconstruction techniquesfor this case. However, if magnification information is present along with shear information,together they give (at least in principle) the tensor magnification which is linear in κ, and so(9) again applies.3Figure 2. At fixed redshift, the mass sheet degeneracy leaves 1 − κuncertain by a multiplicative factor, while the monopole part of κ isundetermined outside the region of the images. As a result, the twomass profiles shown will produce identical multiple-image and sheardata out to the cliff.discussed above, the data enter there either as image positions which are knownvery accurately, or as inequalities which can be set conservatively; so this issuedoes not introduce new difficulties.At least four different possibilities for lens reconstruction now suggest them-selves• Put τ(θ) on a grid and regularize. This may be the simplest approach.• Use basis function expansions for κ(θ) and τ(θ). A Fourier-Bessel expan-sionτ(θ) =∑mncmnJm(kmnθ) exp(imφ)is particularly attractive as it would trivially relate τ and κ.• Pixellate κ(θ) and regularize. This is the one we have implemented. Themass is distributed on square tiles, each having constant but adjustabledensity, and τ(θ) comes from computing Eq. (1) exactly. Note that al-though the mass distribution is discontinuous, the arrival-time surface issmooth. Unlike in the two previous possible approaches, pixellating themass makes it is easy to enforce non-negativity of the mass distribution.• Pixellate κ(θ) and use maximum entropy. See Bridle et al. (this volume)for an example of this in a somewhat different context.The regularization we applied is to minimize∫ (κ− 〈κ〉〈L〉L)2d2θ + ε4∫(∇2κ)2 d2θ (10)while of course enforcing all the lensing constraints. The first term in (10) tendsto minimize mass-to-light variation, since 〈κ〉/〈L〉 is a mean M/L. The secondterm tends to smooth, with ε a sort of smoothing scale. By regularizing withrespect to different light distributions we can get an estimate of the uncertainty.Since the regularizing functional (10) is quadratic in κ, minimizing it sub-ject to the linear lensing constraints is readily implemented through quadratic4Figure 3. Left: Cluster luminosity contours in Cl 1358+62. fromthe catalog of galaxy Vz’s in van Dokkum et al (1998). Likened atthe conference to a left facing seahorse with a ponytail. Right: Shearmap (exaggerated by a factor of 5) from Hoekstra et al (1998). Wehave excised the central region to avoid averaging the shear where itis varying rapidly. Also note that unlike Fig. 14 in Hoekstra et al., wehave binned without smoothing here.programming algorithms. The disadvantage is that the storage needed is '2〈number of pixels〉2 limiting us to of order 5000 pixels. The solution is to haveadaptive pixellation (much like tree codes in N -body simulations), with smallerpixels for the inner parts of the cluster and large pixels outside.4. The mass-sheet degeneracyAfter enthusing about how easy lens reconstruction is, it is well to make acautionary remark about the main source of uncertainty. Here again, the arrival-time surface is very useful.If we multiply τ(θ) by a constant factor, we just stretch the surface verti-cally, and geometrically it is clear that neither image locations nor their relativemagnifications change. More formally, we can rearrange Eq. (2) and drop theirrelevant term 12β2 to getτ(θ) = 2∇−2(1− DlsDsκ)− 12θ · β. (11)If we now multiply both (1−Dls/Dsκ) and β by some constant a, the imagestructure will be unchanged; only time delays and total magnification will getmultiplied by a and a−1 respectively (the latter because rescaling β rescalesall the sources). Therefore lens reconstruction from image structure (withoutabsolute magnifications) leaves (1−Dls/Dsκ) uncertain by a constant factor.This is the mass-sheet degeneracy.5Figure 4. Pixellation used for the mass reconstruction of the Sea-horse cluster. Note that is completely adaptive, and allows us toincrease the resolution at any desired place—in this case the regionaround the red arc. Left: The full field. Right: The inner region,with dots marking features in the red arc.It may appear at first that the mass-sheet degeneracy can be eliminated byrequiring κ to vanish at large distances. But outside the observed region, themonopole part of κ is completely unconstrained. So the mass-sheet degeneracyis equally effective with a mass disk larger than the observed region. Figure 2illustrates.5. Reconstruction of Cl 1358+62 (The Seahorse Cluster)This cluster seemed the obvious test case for combined strong and weak lensingreconstruction in a relatively large field. The cluster itself is at z = 0.33 and itsinner part lenses a z = 4.92 galaxy into a red arc; this was identified by Franxet al. (1997) who also presented a strong lens model. Hoekstra et al. (1998)measured the shear field in an HST WFPC2 mosaic and reconstructed a massmap from weak lensing. The present work combines both regimes.Figure 3 illustrates the data on this cluster that we have used: a shear fieldderived by binning the individual polarization measurements kindly provided byHoekstra and collaborators and assuming a constant z = 1 for the backgroundgalaxies, and a smoothed Vz light distribution. Figure 4 shows the pixellationwe used.Figure 5 is our mass map, computed by regularizing with respect to thelight and a smoothing scale ε changing from ' 5′′ in the center to ' 1′ at theedge. Our estimated uncertainty is derived from an ensemble of reconstructionswhere we rotated the light map by arbitrary angles and shifted it randomly byup to 100′′ and regularized with respect to these altered light maps. The massmap resembles Fig. 15 of Hoekstra et al. (1998) but tends to be smoother. Alsothe overall normalization is somewhat higher; this is probably due to different6Figure 5. Mass map of the Seahorse cluster. Left: Contours of κ insteps of 0.1. Right: Contours of ∆κ in steps of 0.025.treatments of the boundary, though with the mass-sheet degeneracy. (The redarc at z = 4.92 compared to z = 1 assumed for the weakly lensed galaxies,reduces the effect of the mass-sheet degeneracy, but does not eliminate it.) Ourcentral density is higher, but this is as expected since inclusion of a multiply-imaged system immediately forces κ > 1. There is some indication that themass peak is offset (by some 10s of kpc) to the south of the light peak, a naturalthing to expect if the cluster is asymmetric and galaxy formation is biased. Butthis offset is tentative; in Abell 2218, where the inner region is much richer inlensing and better constrained, the evidence for an offset is more compelling(AbdelSalam et al. 1998).Figure 6 shows the enclosed mass out to different radii and the reconstructedshear. It is surprising that the enclosed mass after angular averaging, looks so‘isothermal’ even though the mass map is very asymmetric. We obtain M =(10 ± 1) × 1014M Mpc−1, corresponding to a formal Einstein radius of 43′′and a formal isothermal los dispersion of (990 ± 70) km sec−1. The estimatedmass-to-light is (380 ± 60)hML−1V . The CNOC survey (Carlberg et al. 1997)measured 910 km sec−1 for the los velocity dispersion and estimated M/L =229h−1ML−1V .6. DiscussionWe have developed a mass reconstruction technique that deals with all the clus-ter lensing regimes simultaneously, from multiple images in the central regionsto weak shear in the outer regions, and moreover with adaptive resolution. Codeimplementing this is available from the authors. Variants of our technique (sug-gested in Section 3) may also be of interest in future work.At this stage it appears that cluster mass reconstruction is very good atrecovering features—a good example are offsets between mass and light peaks7Figure 6. Left: Shear map. The superimposed ellipses show the ob-served and reconstructed (dotted) shear, both exaggerated by a factorof five. As we might expect, the reconstructed shear appears more cor-related than the observed. The ellipses in the inner region show thereconstructed shear (not exaggerated) at the locations of the red arc.Right: Enclosed mass and estimated uncertainty.in clusters, which may be indirect evidence for biased galaxy formation. Butthe calibration of the mass (even with perfect shear data) is still problematic—the mass sheet degeneracy is a vicious effect. We suggest that recovering theredshift distribution and the absolute magnification of the background galaxies(even with large uncertainties) are the best hope of breaking this degeneracy,and is likely to be very rewarding in future work.Acknowledgments. We are grateful to Henk Hoekstra and collaboratorsfor supplying us with their shear data on Cl 1358+62.ReferencesAbdelSalam, H.M., Saha, P., & Williams, L.L.R. 1998, AJ, 116, 1541Bridle, S.L., Hobson, M.P., Lasenby, A.N., Saunders, R., this volumeCarlberg, R.G., Yee, H.K.C., Ellingson, E. 1997, ApJ, 478, 462van Dokkum, P.G., Franx, M., Kelson, D.D., Illingworth, G.D., Fisher, D.,Fabricant, D. 1998, ApJ, 500, 714Dye, S., & Taylor, A. 1998, MNRAS, 300, L23Franx, M., Illingworth, G.D., Kelson, D.D., van Dokkum, P.G., Tran, K.-V.1997, ApJ, 486, L75Hoekstra, H., Franx, M. Kuijken, K., Squires, G. 1998 ApJ, 504, 6368

Abstract: The lensing information provided by multiple images, arclets, and statistical distortions can all be formulated as linear constraints on the arrival-time surface, and hence on the mass distribution. This reduces cluster lens reconstruction from combined strong and weak lensing to a standard type of inversion problem. Moreover, the mass sheet degeneracy is broken if there are sources at different redshifts.

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   University of GroningenCluster Reconstruction from Combined Strong and Weak LensingSaha, P.; Williams, L. L. R.; Abdel-Salam, H. M.Published in:ArXivDOI:10.1.1.35.1213IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Saha, P., Williams, L. L. R., & Abdel-Salam, H. M. (1999). Cluster Reconstruction from Combined Strongand Weak Lensing. Manuscript submitted for publication. https://doi.org/10.1.1.35.1213CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 13-02-2023Cluster Reconstruction from Combined Strong and WeakLensingPrasenjit SahaDepartment of Physics, University of Oxford, UKLiliya L.R. WilliamsDepartment of Physics and Astronomy, Univ. of Victoria, BC, CanadaHanadi M. AbdelSalamKapteyn Sterrewacht, Rijksuniversitaet Groningen, The NetherlandsAbstract. The lensing information provided by multiple images, ar-clets, and statistical distortions can all be formulated as linear constraintson the arrival-time surface, and hence on the mass distribution. This re-duces cluster lens reconstruction from combined strong and weak lensingto a standard type of inversion problem. Moreover, the mass sheet de-generacy is broken if there are sources at different redshifts.This paper presents a reconstruction technique based on these ideas,and a mass map and discussion of Cl 1358+62.1. IntroductionThe different regimes of cluster lensing—the inner region where multiple images,including giant arcs are found, further out where highly sheared but singly im-aged arclets are found, and the outer regions showing statistical shear—have inthe past been approached with quite different modeling methods. But these ap-parently separate regimes can be studied in a unified way. The key is to expresslensing information in terms of the arrival surface.In earlier work (see AbdelSalam et al. 1998) we combined the multiple-image and arclet regimes and argued that extending to statistical shear was astraightforward algorithmic issue. This paper makes that extension.2. The arrival-time surfaceThe creature we will mostly be concerned with is the scaled arrival-time surfaceτ(θ), and it is expressed as follows:τ(θ) = 12(θ− β)2 −(Dls/Ds)π∫ln |θ− θ′|κ(θ′) d2θ′ (1)= 12(θ− β)2 − 2(Dls/Ds) ∇−2κ(θ). (2)1θy’θx’                                                                                                  ω θθxyFigure 1. Schematic of an arclet elongated with position angle ω,with (θx′ , θy′) being a coordinate system aligned with the elongation.As usual, θ and β are angular locations on the image and source planes respec-tively, and the D’s are angular diameter distances in units of c/H0. However, κis the convergence for sources at infinity, (hence the factors of Dls/Ds) to makedealing with multiple source redshifts more convenient. The form of Eq. (1) isthen familiar, while Eq. (2) is just a shorthand for the same thing with ∇−2denoting an ‘inverse Laplacian operator’ in two dimensions.To get back to physical arrival time and surface density we usetime = h−1z̃ × 80 days arcsec−2 × τ(θ) (3)surface density = h−1z̃ × 1.2× 1011M arcsec−2 × κ(θ) (4)where z̃ = (1 + zl)Dl but is ' zl.Lensing data constrain the arrival-time surface and hence the mass distri-bution κ(θ) in various ways. We can identify three types of constraints: on thevalues of the arrival time surface at particular points, on the first derivative, andon the second derivative.The first type is when we know the height difference between two or morepoints on the arrival-time surface. This is all-important in time-delay quasarsand H0 measurements, because time delays supply just this height-differenceinformation. But it is not relevant to cluster lensing, at least not yet, thoughwe can always hope for a supernova in a giant arc.The second type of constraint comes from stating Fermat’s principle at eachimage location, i.e.,∇τ(θimage) = 0. (5)Since image positions θimage can generally be measured very accurately, theEq. (5) tells us that the arrival-time surface has zero gradient at some knownpoint. This type of constraint is only useful if we have multiple images; in thatcase we have 2〈images〉 equation of the form (5) but only two unknown sourcecoordinates to solve for, giving us some net constraints on κ(θ). In general thereare 2(〈images〉 − 〈sources〉) constraints.The third type of constraint involves the curvature or second derivative ofthe arrival-time surface, and may come from either arclets or from statisticaldistortions.Consider first a situation like Fig. 1, where an arclet is observed elongatedwith position angle ω. If we are confident that the arclet has been elongated bya factor of at least k, we can consider the coordinate system (θx′ , θy′) aligned2with the elongation and writek∣∣∣∣∣ ∂2∂θ2x′ τ(θimage)∣∣∣∣∣ ≤∣∣∣∣∣ ∂2∂θ2y′ τ(θimage)∣∣∣∣∣ . (6)Further, since ω is known, we can transform Eq. (6) to the unrotated (θx, θy)coordinates. It is not necessary to have highly accurate values of the elongationor its orientation ω—all we have to do is set k conservatively enough that theinequality (6) is valid. The arclet used may itself be part of a multiple-imagesystem; if so we will have to guess the parity to remove the absolute value signsin (6).The same idea can be applied to statistical distortion. In this case the dataprovide an estimate and uncertaintyγi1− κ = gi ±∆gi, (7)where γi and κ are components of the second derivative of τ(θ). We can rewritethis in various ways, such as(1− κ(θimage))gi −∆gi ≤ γ(θimage) ≤ (1− κ(θimage))gi + ∆gi. (8)(We have assumed κ∆gi is negligible here.)A key feature of the constraint equations (5), (6), and (8) is that they areall linear in the unknowns β and κ(θ). Which is to say1( )=( ) ′ (9)and moreover this is true for all lensing regimes: multiple-image, arclet, andstatistical shear.3. Mass reconstructionEq. (9) summarizes the advantage of casting the observational information asconstraints on the arrival-time surface. Lens reconstruction is reduced to a linearinversion problem. It is (as symbolized by the 2 × 5 matrix) a highly underde-termined problem, so in order to reconstruct the lens it is necessary to add extrainformation (sometimes called a prior). Also as indicated in the 2×5 matrix, thedata enter into the linear operator—an unusual complication. However, as we1With one exception: scalar magnification data are quadratic in κ and so (9) does not apply.Taylor and collaborators (see e.g., Dye & Taylor 1998) have developed reconstruction techniquesfor this case. However, if magnification information is present along with shear information,together they give (at least in principle) the tensor magnification which is linear in κ, and so(9) again applies.3Figure 2. At fixed redshift, the mass sheet degeneracy leaves 1 − κuncertain by a multiplicative factor, while the monopole part of κ isundetermined outside the region of the images. As a result, the twomass profiles shown will produce identical multiple-image and sheardata out to the cliff.discussed above, the data enter there either as image positions which are knownvery accurately, or as inequalities which can be set conservatively; so this issuedoes not introduce new difficulties.At least four different possibilities for lens reconstruction now suggest them-selves• Put τ(θ) on a grid and regularize. This may be the simplest approach.• Use basis function expansions for κ(θ) and τ(θ). A Fourier-Bessel expan-sionτ(θ) =∑mncmnJm(kmnθ) exp(imφ)is particularly attractive as it would trivially relate τ and κ.• Pixellate κ(θ) and regularize. This is the one we have implemented. Themass is distributed on square tiles, each having constant but adjustabledensity, and τ(θ) comes from computing Eq. (1) exactly. Note that al-though the mass distribution is discontinuous, the arrival-time surface issmooth. Unlike in the two previous possible approaches, pixellating themass makes it is easy to enforce non-negativity of the mass distribution.• Pixellate κ(θ) and use maximum entropy. See Bridle et al. (this volume)for an example of this in a somewhat different context.The regularization we applied is to minimize∫ (κ− 〈κ〉〈L〉L)2d2θ + ε4∫(∇2κ)2 d2θ (10)while of course enforcing all the lensing constraints. The first term in (10) tendsto minimize mass-to-light variation, since 〈κ〉/〈L〉 is a mean M/L. The secondterm tends to smooth, with ε a sort of smoothing scale. By regularizing withrespect to different light distributions we can get an estimate of the uncertainty.Since the regularizing functional (10) is quadratic in κ, minimizing it sub-ject to the linear lensing constraints is readily implemented through quadratic4Figure 3. Left: Cluster luminosity contours in Cl 1358+62. fromthe catalog of galaxy Vz’s in van Dokkum et al (1998). Likened atthe conference to a left facing seahorse with a ponytail. Right: Shearmap (exaggerated by a factor of 5) from Hoekstra et al (1998). Wehave excised the central region to avoid averaging the shear where itis varying rapidly. Also note that unlike Fig. 14 in Hoekstra et al., wehave binned without smoothing here.programming algorithms. The disadvantage is that the storage needed is '2〈number of pixels〉2 limiting us to of order 5000 pixels. The solution is to haveadaptive pixellation (much like tree codes in N -body simulations), with smallerpixels for the inner parts of the cluster and large pixels outside.4. The mass-sheet degeneracyAfter enthusing about how easy lens reconstruction is, it is well to make acautionary remark about the main source of uncertainty. Here again, the arrival-time surface is very useful.If we multiply τ(θ) by a constant factor, we just stretch the surface verti-cally, and geometrically it is clear that neither image locations nor their relativemagnifications change. More formally, we can rearrange Eq. (2) and drop theirrelevant term 12β2 to getτ(θ) = 2∇−2(1− DlsDsκ)− 12θ · β. (11)If we now multiply both (1−Dls/Dsκ) and β by some constant a, the imagestructure will be unchanged; only time delays and total magnification will getmultiplied by a and a−1 respectively (the latter because rescaling β rescalesall the sources). Therefore lens reconstruction from image structure (withoutabsolute magnifications) leaves (1−Dls/Dsκ) uncertain by a constant factor.This is the mass-sheet degeneracy.5Figure 4. Pixellation used for the mass reconstruction of the Sea-horse cluster. Note that is completely adaptive, and allows us toincrease the resolution at any desired place—in this case the regionaround the red arc. Left: The full field. Right: The inner region,with dots marking features in the red arc.It may appear at first that the mass-sheet degeneracy can be eliminated byrequiring κ to vanish at large distances. But outside the observed region, themonopole part of κ is completely unconstrained. So the mass-sheet degeneracyis equally effective with a mass disk larger than the observed region. Figure 2illustrates.5. Reconstruction of Cl 1358+62 (The Seahorse Cluster)This cluster seemed the obvious test case for combined strong and weak lensingreconstruction in a relatively large field. The cluster itself is at z = 0.33 and itsinner part lenses a z = 4.92 galaxy into a red arc; this was identified by Franxet al. (1997) who also presented a strong lens model. Hoekstra et al. (1998)measured the shear field in an HST WFPC2 mosaic and reconstructed a massmap from weak lensing. The present work combines both regimes.Figure 3 illustrates the data on this cluster that we have used: a shear fieldderived by binning the individual polarization measurements kindly provided byHoekstra and collaborators and assuming a constant z = 1 for the backgroundgalaxies, and a smoothed Vz light distribution. Figure 4 shows the pixellationwe used.Figure 5 is our mass map, computed by regularizing with respect to thelight and a smoothing scale ε changing from ' 5′′ in the center to ' 1′ at theedge. Our estimated uncertainty is derived from an ensemble of reconstructionswhere we rotated the light map by arbitrary angles and shifted it randomly byup to 100′′ and regularized with respect to these altered light maps. The massmap resembles Fig. 15 of Hoekstra et al. (1998) but tends to be smoother. Alsothe overall normalization is somewhat higher; this is probably due to different6Figure 5. Mass map of the Seahorse cluster. Left: Contours of κ insteps of 0.1. Right: Contours of ∆κ in steps of 0.025.treatments of the boundary, though with the mass-sheet degeneracy. (The redarc at z = 4.92 compared to z = 1 assumed for the weakly lensed galaxies,reduces the effect of the mass-sheet degeneracy, but does not eliminate it.) Ourcentral density is higher, but this is as expected since inclusion of a multiply-imaged system immediately forces κ > 1. There is some indication that themass peak is offset (by some 10s of kpc) to the south of the light peak, a naturalthing to expect if the cluster is asymmetric and galaxy formation is biased. Butthis offset is tentative; in Abell 2218, where the inner region is much richer inlensing and better constrained, the evidence for an offset is more compelling(AbdelSalam et al. 1998).Figure 6 shows the enclosed mass out to different radii and the reconstructedshear. It is surprising that the enclosed mass after angular averaging, looks so‘isothermal’ even though the mass map is very asymmetric. We obtain M =(10 ± 1) × 1014M Mpc−1, corresponding to a formal Einstein radius of 43′′and a formal isothermal los dispersion of (990 ± 70) km sec−1. The estimatedmass-to-light is (380 ± 60)hML−1V . The CNOC survey (Carlberg et al. 1997)measured 910 km sec−1 for the los velocity dispersion and estimated M/L =229h−1ML−1V .6. DiscussionWe have developed a mass reconstruction technique that deals with all the clus-ter lensing regimes simultaneously, from multiple images in the central regionsto weak shear in the outer regions, and moreover with adaptive resolution. Codeimplementing this is available from the authors. Variants of our technique (sug-gested in Section 3) may also be of interest in future work.At this stage it appears that cluster mass reconstruction is very good atrecovering features—a good example are offsets between mass and light peaks7Figure 6. Left: Shear map. The superimposed ellipses show the ob-served and reconstructed (dotted) shear, both exaggerated by a factorof five. As we might expect, the reconstructed shear appears more cor-related than the observed. The ellipses in the inner region show thereconstructed shear (not exaggerated) at the locations of the red arc.Right: Enclosed mass and estimated uncertainty.in clusters, which may be indirect evidence for biased galaxy formation. Butthe calibration of the mass (even with perfect shear data) is still problematic—the mass sheet degeneracy is a vicious effect. We suggest that recovering theredshift distribution and the absolute magnification of the background galaxies(even with large uncertainties) are the best hope of breaking this degeneracy,and is likely to be very rewarding in future work.Acknowledgments. We are grateful to Henk Hoekstra and collaboratorsfor supplying us with their shear data on Cl 1358+62.ReferencesAbdelSalam, H.M., Saha, P., & Williams, L.L.R. 1998, AJ, 116, 1541Bridle, S.L., Hobson, M.P., Lasenby, A.N., Saunders, R., this volumeCarlberg, R.G., Yee, H.K.C., Ellingson, E. 1997, ApJ, 478, 462van Dokkum, P.G., Franx, M., Kelson, D.D., Illingworth, G.D., Fisher, D.,Fabricant, D. 1998, ApJ, 500, 714Dye, S., & Taylor, A. 1998, MNRAS, 300, L23Franx, M., Illingworth, G.D., Kelson, D.D., van Dokkum, P.G., Tran, K.-V.1997, ApJ, 486, L75Hoekstra, H., Franx, M. Kuijken, K., Squires, G. 1998 ApJ, 504, 6368

   University of GroningenCluster Reconstruction from Combined Strong and Weak LensingSaha, P.; Williams, L. L. R.; Abdel-Salam, H. M.Published in:ArXivDOI:10.1.1.35.1213IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Saha, P., Williams, L. L. R., & Abdel-Salam, H. M. (1999). Cluster Reconstruction from Combined Strongand Weak Lensing. Manuscript submitted for publication. https://doi.org/10.1.1.35.1213CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 09-09-2023Cluster Reconstruction from Combined Strong and WeakLensingPrasenjit SahaDepartment of Physics, University of Oxford, UKLiliya L.R. WilliamsDepartment of Physics and Astronomy, Univ. of Victoria, BC, CanadaHanadi M. AbdelSalamKapteyn Sterrewacht, Rijksuniversitaet Groningen, The NetherlandsAbstract. The lensing information provided by multiple images, ar-clets, and statistical distortions can all be formulated as linear constraintson the arrival-time surface, and hence on the mass distribution. This re-duces cluster lens reconstruction from combined strong and weak lensingto a standard type of inversion problem. Moreover, the mass sheet de-generacy is broken if there are sources at different redshifts.This paper presents a reconstruction technique based on these ideas,and a mass map and discussion of Cl 1358+62.1. IntroductionThe different regimes of cluster lensing—the inner region where multiple images,including giant arcs are found, further out where highly sheared but singly im-aged arclets are found, and the outer regions showing statistical shear—have inthe past been approached with quite different modeling methods. But these ap-parently separate regimes can be studied in a unified way. The key is to expresslensing information in terms of the arrival surface.In earlier work (see AbdelSalam et al. 1998) we combined the multiple-image and arclet regimes and argued that extending to statistical shear was astraightforward algorithmic issue. This paper makes that extension.2. The arrival-time surfaceThe creature we will mostly be concerned with is the scaled arrival-time surfaceτ(θ), and it is expressed as follows:τ(θ) = 12(θ− β)2 −(Dls/Ds)π∫ln |θ− θ′|κ(θ′) d2θ′ (1)= 12(θ− β)2 − 2(Dls/Ds) ∇−2κ(θ). (2)1θy’θx’                                                                                                  ω θθxyFigure 1. Schematic of an arclet elongated with position angle ω,with (θx′ , θy′) being a coordinate system aligned with the elongation.As usual, θ and β are angular locations on the image and source planes respec-tively, and the D’s are angular diameter distances in units of c/H0. However, κis the convergence for sources at infinity, (hence the factors of Dls/Ds) to makedealing with multiple source redshifts more convenient. The form of Eq. (1) isthen familiar, while Eq. (2) is just a shorthand for the same thing with ∇−2denoting an ‘inverse Laplacian operator’ in two dimensions.To get back to physical arrival time and surface density we usetime = h−1z̃ × 80 days arcsec−2 × τ(θ) (3)surface density = h−1z̃ × 1.2× 1011M arcsec−2 × κ(θ) (4)where z̃ = (1 + zl)Dl but is ' zl.Lensing data constrain the arrival-time surface and hence the mass distri-bution κ(θ) in various ways. We can identify three types of constraints: on thevalues of the arrival time surface at particular points, on the first derivative, andon the second derivative.The first type is when we know the height difference between two or morepoints on the arrival-time surface. This is all-important in time-delay quasarsand H0 measurements, because time delays supply just this height-differenceinformation. But it is not relevant to cluster lensing, at least not yet, thoughwe can always hope for a supernova in a giant arc.The second type of constraint comes from stating Fermat’s principle at eachimage location, i.e.,∇τ(θimage) = 0. (5)Since image positions θimage can generally be measured very accurately, theEq. (5) tells us that the arrival-time surface has zero gradient at some knownpoint. This type of constraint is only useful if we have multiple images; in thatcase we have 2〈images〉 equation of the form (5) but only two unknown sourcecoordinates to solve for, giving us some net constraints on κ(θ). In general thereare 2(〈images〉 − 〈sources〉) constraints.The third type of constraint involves the curvature or second derivative ofthe arrival-time surface, and may come from either arclets or from statisticaldistortions.Consider first a situation like Fig. 1, where an arclet is observed elongatedwith position angle ω. If we are confident that the arclet has been elongated bya factor of at least k, we can consider the coordinate system (θx′ , θy′) aligned2with the elongation and writek∣∣∣∣∣ ∂2∂θ2x′ τ(θimage)∣∣∣∣∣ ≤∣∣∣∣∣ ∂2∂θ2y′ τ(θimage)∣∣∣∣∣ . (6)Further, since ω is known, we can transform Eq. (6) to the unrotated (θx, θy)coordinates. It is not necessary to have highly accurate values of the elongationor its orientation ω—all we have to do is set k conservatively enough that theinequality (6) is valid. The arclet used may itself be part of a multiple-imagesystem; if so we will have to guess the parity to remove the absolute value signsin (6).The same idea can be applied to statistical distortion. In this case the dataprovide an estimate and uncertaintyγi1− κ = gi ±∆gi, (7)where γi and κ are components of the second derivative of τ(θ). We can rewritethis in various ways, such as(1− κ(θimage))gi −∆gi ≤ γ(θimage) ≤ (1− κ(θimage))gi + ∆gi. (8)(We have assumed κ∆gi is negligible here.)A key feature of the constraint equations (5), (6), and (8) is that they areall linear in the unknowns β and κ(θ). Which is to say1( )=( ) ′ (9)and moreover this is true for all lensing regimes: multiple-image, arclet, andstatistical shear.3. Mass reconstructionEq. (9) summarizes the advantage of casting the observational information asconstraints on the arrival-time surface. Lens reconstruction is reduced to a linearinversion problem. It is (as symbolized by the 2 × 5 matrix) a highly underde-termined problem, so in order to reconstruct the lens it is necessary to add extrainformation (sometimes called a prior). Also as indicated in the 2×5 matrix, thedata enter into the linear operator—an unusual complication. However, as we1With one exception: scalar magnification data are quadratic in κ and so (9) does not apply.Taylor and collaborators (see e.g., Dye & Taylor 1998) have developed reconstruction techniquesfor this case. However, if magnification information is present along with shear information,together they give (at least in principle) the tensor magnification which is linear in κ, and so(9) again applies.3Figure 2. At fixed redshift, the mass sheet degeneracy leaves 1 − κuncertain by a multiplicative factor, while the monopole part of κ isundetermined outside the region of the images. As a result, the twomass profiles shown will produce identical multiple-image and sheardata out to the cliff.discussed above, the data enter there either as image positions which are knownvery accurately, or as inequalities which can be set conservatively; so this issuedoes not introduce new difficulties.At least four different possibilities for lens reconstruction now suggest them-selves• Put τ(θ) on a grid and regularize. This may be the simplest approach.• Use basis function expansions for κ(θ) and τ(θ). A Fourier-Bessel expan-sionτ(θ) =∑mncmnJm(kmnθ) exp(imφ)is particularly attractive as it would trivially relate τ and κ.• Pixellate κ(θ) and regularize. This is the one we have implemented. Themass is distributed on square tiles, each having constant but adjustabledensity, and τ(θ) comes from computing Eq. (1) exactly. Note that al-though the mass distribution is discontinuous, the arrival-time surface issmooth. Unlike in the two previous possible approaches, pixellating themass makes it is easy to enforce non-negativity of the mass distribution.• Pixellate κ(θ) and use maximum entropy. See Bridle et al. (this volume)for an example of this in a somewhat different context.The regularization we applied is to minimize∫ (κ− 〈κ〉〈L〉L)2d2θ + ε4∫(∇2κ)2 d2θ (10)while of course enforcing all the lensing constraints. The first term in (10) tendsto minimize mass-to-light variation, since 〈κ〉/〈L〉 is a mean M/L. The secondterm tends to smooth, with ε a sort of smoothing scale. By regularizing withrespect to different light distributions we can get an estimate of the uncertainty.Since the regularizing functional (10) is quadratic in κ, minimizing it sub-ject to the linear lensing constraints is readily implemented through quadratic4Figure 3. Left: Cluster luminosity contours in Cl 1358+62. fromthe catalog of galaxy Vz’s in van Dokkum et al (1998). Likened atthe conference to a left facing seahorse with a ponytail. Right: Shearmap (exaggerated by a factor of 5) from Hoekstra et al (1998). Wehave excised the central region to avoid averaging the shear where itis varying rapidly. Also note that unlike Fig. 14 in Hoekstra et al., wehave binned without smoothing here.programming algorithms. The disadvantage is that the storage needed is '2〈number of pixels〉2 limiting us to of order 5000 pixels. The solution is to haveadaptive pixellation (much like tree codes in N -body simulations), with smallerpixels for the inner parts of the cluster and large pixels outside.4. The mass-sheet degeneracyAfter enthusing about how easy lens reconstruction is, it is well to make acautionary remark about the main source of uncertainty. Here again, the arrival-time surface is very useful.If we multiply τ(θ) by a constant factor, we just stretch the surface verti-cally, and geometrically it is clear that neither image locations nor their relativemagnifications change. More formally, we can rearrange Eq. (2) and drop theirrelevant term 12β2 to getτ(θ) = 2∇−2(1− DlsDsκ)− 12θ · β. (11)If we now multiply both (1−Dls/Dsκ) and β by some constant a, the imagestructure will be unchanged; only time delays and total magnification will getmultiplied by a and a−1 respectively (the latter because rescaling β rescalesall the sources). Therefore lens reconstruction from image structure (withoutabsolute magnifications) leaves (1−Dls/Dsκ) uncertain by a constant factor.This is the mass-sheet degeneracy.5Figure 4. Pixellation used for the mass reconstruction of the Sea-horse cluster. Note that is completely adaptive, and allows us toincrease the resolution at any desired place—in this case the regionaround the red arc. Left: The full field. Right: The inner region,with dots marking features in the red arc.It may appear at first that the mass-sheet degeneracy can be eliminated byrequiring κ to vanish at large distances. But outside the observed region, themonopole part of κ is completely unconstrained. So the mass-sheet degeneracyis equally effective with a mass disk larger than the observed region. Figure 2illustrates.5. Reconstruction of Cl 1358+62 (The Seahorse Cluster)This cluster seemed the obvious test case for combined strong and weak lensingreconstruction in a relatively large field. The cluster itself is at z = 0.33 and itsinner part lenses a z = 4.92 galaxy into a red arc; this was identified by Franxet al. (1997) who also presented a strong lens model. Hoekstra et al. (1998)measured the shear field in an HST WFPC2 mosaic and reconstructed a massmap from weak lensing. The present work combines both regimes.Figure 3 illustrates the data on this cluster that we have used: a shear fieldderived by binning the individual polarization measurements kindly provided byHoekstra and collaborators and assuming a constant z = 1 for the backgroundgalaxies, and a smoothed Vz light distribution. Figure 4 shows the pixellationwe used.Figure 5 is our mass map, computed by regularizing with respect to thelight and a smoothing scale ε changing from ' 5′′ in the center to ' 1′ at theedge. Our estimated uncertainty is derived from an ensemble of reconstructionswhere we rotated the light map by arbitrary angles and shifted it randomly byup to 100′′ and regularized with respect to these altered light maps. The massmap resembles Fig. 15 of Hoekstra et al. (1998) but tends to be smoother. Alsothe overall normalization is somewhat higher; this is probably due to different6Figure 5. Mass map of the Seahorse cluster. Left: Contours of κ insteps of 0.1. Right: Contours of ∆κ in steps of 0.025.treatments of the boundary, though with the mass-sheet degeneracy. (The redarc at z = 4.92 compared to z = 1 assumed for the weakly lensed galaxies,reduces the effect of the mass-sheet degeneracy, but does not eliminate it.) Ourcentral density is higher, but this is as expected since inclusion of a multiply-imaged system immediately forces κ > 1. There is some indication that themass peak is offset (by some 10s of kpc) to the south of the light peak, a naturalthing to expect if the cluster is asymmetric and galaxy formation is biased. Butthis offset is tentative; in Abell 2218, where the inner region is much richer inlensing and better constrained, the evidence for an offset is more compelling(AbdelSalam et al. 1998).Figure 6 shows the enclosed mass out to different radii and the reconstructedshear. It is surprising that the enclosed mass after angular averaging, looks so‘isothermal’ even though the mass map is very asymmetric. We obtain M =(10 ± 1) × 1014M Mpc−1, corresponding to a formal Einstein radius of 43′′and a formal isothermal los dispersion of (990 ± 70) km sec−1. The estimatedmass-to-light is (380 ± 60)hML−1V . The CNOC survey (Carlberg et al. 1997)measured 910 km sec−1 for the los velocity dispersion and estimated M/L =229h−1ML−1V .6. DiscussionWe have developed a mass reconstruction technique that deals with all the clus-ter lensing regimes simultaneously, from multiple images in the central regionsto weak shear in the outer regions, and moreover with adaptive resolution. Codeimplementing this is available from the authors. Variants of our technique (sug-gested in Section 3) may also be of interest in future work.At this stage it appears that cluster mass reconstruction is very good atrecovering features—a good example are offsets between mass and light peaks7Figure 6. Left: Shear map. The superimposed ellipses show the ob-served and reconstructed (dotted) shear, both exaggerated by a factorof five. As we might expect, the reconstructed shear appears more cor-related than the observed. The ellipses in the inner region show thereconstructed shear (not exaggerated) at the locations of the red arc.Right: Enclosed mass and estimated uncertainty.in clusters, which may be indirect evidence for biased galaxy formation. Butthe calibration of the mass (even with perfect shear data) is still problematic—the mass sheet degeneracy is a vicious effect. We suggest that recovering theredshift distribution and the absolute magnification of the background galaxies(even with large uncertainties) are the best hope of breaking this degeneracy,and is likely to be very rewarding in future work.Acknowledgments. We are grateful to Henk Hoekstra and collaboratorsfor supplying us with their shear data on Cl 1358+62.ReferencesAbdelSalam, H.M., Saha, P., & Williams, L.L.R. 1998, AJ, 116, 1541Bridle, S.L., Hobson, M.P., Lasenby, A.N., Saunders, R., this volumeCarlberg, R.G., Yee, H.K.C., Ellingson, E. 1997, ApJ, 478, 462van Dokkum, P.G., Franx, M., Kelson, D.D., Illingworth, G.D., Fisher, D.,Fabricant, D. 1998, ApJ, 500, 714Dye, S., & Taylor, A. 1998, MNRAS, 300, L23Franx, M., Illingworth, G.D., Kelson, D.D., van Dokkum, P.G., Tran, K.-V.1997, ApJ, 486, L75Hoekstra, H., Franx, M. Kuijken, K., Squires, G. 1998 ApJ, 504, 6368

Cluster Reconstruction from Combined Strong and Weak Lensing

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