2. Outline Brief, non-technical introduction to strong (multiple image) lensing Bayesian approach to the reconstruction of lens mass distribution Overview of mass reconstruction methods and results Non-parametric (free-form) lens reconstruction method: PixeLens Open questions and future work Galaxy cluster Abell 1689

3. A Brief Introduction to Lensing total travel time position on the sky Goal: Goal: find positions of images on the plane of the sky How? How? use Fermats Principle - images are formed at the local minima, maxima and saddle points of the total light travel time (arrival time) from source to observer

4. A Brief Introduction to Lensing Circularly symmetric lens On-axis source Circularly symmetric lens Off-axis source Elliptical lens Off-axis source Plane of the sky

5. All the Information about Images is contained in the Arrival Time Surface Positions: Images form at the extrema, or stationary points (minima, maxima, saddles) of the arrival time surface. Time Delays: A light pulse from the source will arrive at the observer at 5 different times: the time delays between images are equal to the difference in the height of the arrival time surface. Magnifications: The magnification and distortion, or shearing of images is given by the curvature of the arrival time surface. [Schneider 1985] [Blandford & Narayan 1986]

6. Substructure and Image Properties smooth elliptical lens … with mass lump (~1%) added Maxima, minima, saddles of the arrival time surface correspond to images

7. Examples of Lens Systems ~ 1 arcminute ~1 arcsecond Galaxy Clusters Galaxies Properties of lensed images provide precise information about the total (dark and light) mass distribution can get dark matter mass map. Clumping properties of dark matter the nature of dark matter particles. We would like to reconstruct mass distribution without any regard to how light is distributed.

8. Bayesian approach to lens mass reconstruction #data > #model parameters P(D|H,I) dominates P(H|I) not important #data < #model parameters P(H|I) is important ! D is data with errors P(D|H,I) is the usual -type fcn P(H|I) provides regularization parametric methods 5-10 parameters P(H|I) choices: maximum entropy min. w.r.t. observed light smoothing (local, global) … D is exact (perfect data) P(D|H,I) is replaced by linear constraints P(H|I): can use additional constraints P(H|I) can also provide regularization D is exact (perfect data) P(D|H,I) is replaced by linear constraints P(H|I) is replaced by linear constraints no regularization -> ensemble average prior likelihood evidence PixeLens posterior

9. Parametric–unknowns: Parametric – unknowns: masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a 2 -type function data without errors: P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior (MaxEnt; minimize w.r.t light; smoothing) linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization – several possibilities if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005) Mass Modeling Methods

10. Question: what is the size of cluster galaxies? Each galaxys mass, radius are fcn (Lum) galaxy + cluster mass are superimposed Abell 2218, z=0.175 Within 1 Mpc of cluster center galaxies comprise 10-20% of mass; collisionless DM consistent with collisionless DM collisional fluid-like DM predictions collisionless DM predictions Best fit to 25 galaxies 520 kpc Parametric mass reconstruction: Kneib et al. (1996), Natarajan et al. (2002) Maximize P(D|H,I) likelihood fcn

11. Parametric–unknowns: Parametric – unknowns: masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a 2 -type function data without errors: P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior: minimize w.r.t light; smoothing linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: dispersion bet. scrambled light reconstructions if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005) Mass Modeling Methods

12. Cluster Abell 2218 (z=0.175) Free-form mass reconstruction with regularization: AbdelSalam et al. (1998) 260 kpc Lens eqn is linear in the unknowns Lens eqn is linear in the unknowns: mass pixels, source positions Image elongations also provide linear constraints. Data: Data: coords, elongations of 9 images (4 sources) & 18 arclets Pixelate mass distribution Pixelate mass distribution ~ 3000 pixels (unknowns) Regularize Regularize w.r.t. light distribution Errors: Errors: rms of mass maps with randomized light distribution P(D|H,I) replaced by linear constraints P(H|I)

13. Mass/Light ratios of 3 galaxies differ by x 10 Free-form mass reconstruction with regularization: AbdelSalam et al. (1998) Overall, mass distribution follows light, but: center of mass center of light are displaced by ~ 30 kpc (~ 3 x Suns dist. from Milky Ways center) Chandra Chandra X-ray emission elongated horizontally; X-ray peak close to the predicted mass peak. Machacek et al. (2002) Cluster Abell 2218 (z=0.175) centroid peak

14. Free-form mass reconstruction with regularization: AbdelSalam et al. (1997) Cluster Abell 370 (z=0.375) Color map: Color map: optical image of the cluster Contours: Contours: recovered surface density map Regularized w.r.t. observed light image Regularized w.r.t. a flat light image

15. Free-form mass reconstruction with regularization: AbdelSalam et al. (1997) Cluster Abell 370 (z=0.375) Contours of constant fractional error in the recovered surface density

16. Parametric–unknowns: Parametric – unknowns: masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a 2 -type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior: smoothing linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: bootstrap resampling of data if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005) Mass Modeling Methods

17. Known mass distribution: Known mass distribution: N-body cluster Free-form potential reconstruction with regularization: Bradac et al. (2005a) Solve for the potential Solve for the potential on a grid: 20x20 50x50Minimize: Error estimation Error estimation: bootstrap resampling of weakly lensed galaxies Reconstructions: Reconstructions: starting from three input maps; using 210 arclets, 1 four-image system likelihood moving prior regularization

18. Cluster RX J1347.5-1145 (z=0.451) Free-form potential reconstruction with regularization: Bradac et al. (2005b) Reconstructions: Reconstructions: starting from three input maps; using 210 arclets, 1 three-image system Essentially, weak lensing reconstruction with one multiple image system to break mass sheet degeneracy Cluster mass, r<0.5 Mpc = 1.3 Mpc

19. Parametric–unknowns: Parametric – unknowns: masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a 2 -type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables; adaptive pixel size greater than # observables what prior P(H|I) to use? regularization prior: source size linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization: the intrinsic size of lensed sources is specified if ensemble average – dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005) Mass Modeling Methods

20. Contours: input mass contours Gray scale: recovered mass Free-form mass reconstruction with regularization: Diego et al. (2005b) Known mass distribution: 1 large + 3 small NFW profiles Lens equations: N = [N x M matrix] M N – image positions M – unknowns: mass pixels, source pos. Pixelate mass Pixelate mass: start with ~12 x 12 grid, end up with ~500 pixels in a multi-resolution grid.Sources: extended, few pixels each Minimize R 2 Minimize R 2 : R = N – [N x M] M; residuals vectorInputs: Prior R 2 Initial guess for M unknowns P(H|I) P(D|H,I) replaced by linear constraints

21. Abell 1689, z=0.183 106 images from 30 sources [Broadhurst et al. 2005]

22. Free-form mass reconstruction with regularization: Diego et al. (2005b) Cluster Abell 1689 (z=0.183) 1 arcmin 185 kpc contour lines: reconstructed mass distribution map of S/N ratios Data: Data: 106 images (30 sources) but 601 data pixels Mass pixels Mass pixels: 600, variable size Errors Errors: rms of many reconstructions using different initial conditions (pixel masses, source positions, source redshifts – within error)

23. Parametric–unknowns: Parametric – unknowns: masses, ellipticities, etc. of individual galaxies sufficient for some purposes, but not general enough Kneib et al. (1996), Natarajan et al. (2002), Broadhurst et al. (2004) Free-form – unknowns: Free-form – unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)? lensing potential – automatically accounts for external shear mass – ensures mass non-negativity what data and errors to use? strong lensing (multiply imaged sources), weak lensing (singly imaged) data with errors: P(D|H,I) is usually a 2 -type function data without errors (perfect data): P(D|H,I) replaced by linear constraints how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use? regularization prior (MaxEnt; minimize w.r.t light; smoothing) linear constraints motivated by knowledge of galaxies, clusters how to estimate errors? if regularization – several possibilities if ensemble average: dispersion between individual models AbdelSalam et al. (1997,98), Bradac et al. (2005a,b), Diego et al. (2005a,b) PixeLens: Saha & Williams (2004), Williams & Saha (2005) Mass Modeling Methods

24. Blue – true mass contours Black – reconstructed Red – images of point sources Solve for mass: ~30 x 30 grid of mass pixelsData: P(D|H,I) replaced by linear constraints from image pos. Priors P(H|I): mass pixels non-negative lens center known density gradient must point within of radial -0.1 < 2D density slope < -3 (no smoothness constraint) Ensemble average: 200 models, each reproduces image positions exactly. Free-form mass reconstruction with ensemble averaging: PixeLens Known mass distribution 5 images (1 source) 13 images (3 sources)

25. [Oguri et al. 2004] [Inada et al. 2003, 2005] [Williams & Saha 2005] 15 115 kpc SDSS J1004, z QSO = 1.734 blue crosses: galaxies (not used in modeling) red dots: QSO images Fixed constraints: Fixed constraints: positions of 4 QSO images Priors: external shear PA = 10 45 deg. (Oguri et al. 2004) -0.25 < 2D density slope < -3.0 density gradient direction constraint: must point within 45 or 8 deg. from radial Free-form mass reconstruction with ensemble averaging: PixeLens

26. [Oguri et al. 2004] [Inada et al. 2003, 2005] [Williams & Saha 2005] 15 115 kpc SDSS J1004, z QSO = 1.734 contours: …-6.25, -3.15, 0, 3.15, 6.25… x 10 9 M Sun /arcsec 2 dashed solid 19 galaxies within 120 kpc of cluster center: comprise <10% of mass, have 3<Mass/Light<15 galaxies were stripped of their DM blue crosses: galaxies (not used in modeling) red dots: QSO images Mass maps of residuals for 2 PixeLens reconstructions density slope -1.25 density slope -0.39

27. Conclusions PixeLens PixeLens – easy to use, open source lens modeling code, with a GUI interface (Saha & Williams 2004); use to find it. Galaxy clusters: In general, mass follows light Galaxies within ~20% of the virial radius are stripped of their DM Unrelaxed clusters: mass peak may not coincide with the cD galaxy Results consistent with the predictions of cold dark matter cosmologies Mass reconstruction methods: Parametric models sufficient for some purposes, but to allow for substructure, galaxies variable Mass/Light ratios, misaligned mass/light peaks, and other surprises need more flexible, free-form modeling Open questions in free-form reconstructions: Influence of priors – investigate using reconstructions of synthetic lenses Reducing number of parameters: adaptive pixel size/resolution Principal Components Analysis How to avoid spatially uneven noise distribution in the recovered maps