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

A Brief Introduction to Lensing

total travel time

posi

tion

on

th

e s

ky

Goal:Goal: find positions of images on the plane of the sky

How?How? use Fermat’s Principle - images are formed at the local minima, maxima and saddle points of the total light travel time (arrival time) from source to observer

A Brief Introduction to Lensing

Circularly symmetric lensOn-axis source

Circularly symmetric lensOff-axis source

Elliptical lensOff-axis source

Plane of the sky

All the Information about Imagesis contained in the Arrival Time

SurfacePositions: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 arriveat the observer at 5 different times:the time delays between images areequal to the difference in the “height”of the arrival time surface.

Magnifications:The magnification and distortion, orshearing of images is given by the curvature of the arrival time surface.

[Schneider 1985][Blandford & Narayan 1986]

Substructure and Image Properties

smooth elliptical lens … with mass lump (~1%) added

Maxima, minima, saddles of the arrival time surface correspond to imagesMaxima, minima, saddles of the arrival time surface correspond to images

Examples of Lens Systems

~ 1 arcminute~1 arcsecondGalaxy ClustersGalaxies

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.

Bayesian approach to lens mass reconstruction

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IHDPIHPIDHP

#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 fcnP(H|I) provides regularization

parametric methods5-10 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 constraintsP(H|I): can use additional constraintsP(H|I) can also provide regularization

D is exact (perfect data)P(D|H,I) is replaced by linear constraintsP(H|I) is replaced by linear constraintsno regularization -> ensemble average

prior likelihood

evidence

Pix

eLen

s

posterior

ParametricParametric –– unknowns: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 – Free-form – unknowns:unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)?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?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?how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use?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?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

Question: Question: what is the size of cluster galaxies?what is the size of cluster galaxies?Each galaxy’s mass, radius are fcn (Lum)galaxy + cluster mass are superimposed

Abell 2218,z=0.175

Within 1 Mpc of cluster centergalaxies comprise 10-20% of mass;consistent with collisionless DMcollisionless DM

collisionalfluid-like DMpredictions

collisionlessDM 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

ParametricParametric –– unknowns: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 – Free-form – unknowns:unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)?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?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?how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use?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?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

Cluster Abell 2218 (z=0.175)Cluster Abell 2218 (z=0.175)

Free-form mass reconstruction with

regularization: AbdelSalam et al. (1998)

26

0 k

pc

Lens eqn is linear in the unknownsLens eqn is linear in the unknowns: mass pixels, source positions

Image elongations also provide linear constraints.Image elongations also provide linear constraints.Data:Data: coords, elongations of 9 images (4 sources) & 18 arclets Pixelate mass distributionPixelate mass distribution ~ 3000 pixels (unknowns)RegularizeRegularize w.r.t. light distributionErrors:Errors: rms of mass maps with randomized light distribution

)( impixpiximsrc f

P(D|H,I)replaced bylinear constraints

P(H|I)

Mass/Light ratios of 3 galaxiesdiffer by x 10

Free-form mass reconstruction with

regularization: AbdelSalam et al. (1998)

Overall, Overall, mass distribution mass distribution follows light, but:follows light, but:

center of mass center of light are displacedby ~ 30 kpc(~ 3 x Sun’s dist.from Milky Way’s center)ChandraChandra X-ray emission elongated “horizontally”;

X-ray peak close to the predicted mass peak.

Machacek et al. (2002)

Cluster Abell 2218 (z=0.175)Cluster Abell 2218 (z=0.175)

centr

oid

peak

Free-form mass reconstruction with

regularization: AbdelSalam et al. (1997)Cluster Abell 370 (z=0.375)Cluster Abell 370 (z=0.375) Color map:Color map: optical image of the cluster

Contours:Contours: recovered surface density mapRegularized w.r.t. observed light image Regularized w.r.t. a flat “light” imageRegularized w.r.t. observed light image Regularized w.r.t. a flat “light” image

Free-form mass reconstruction with

regularization: AbdelSalam et al. (1997)

Cluster Abell 370 (z=0.375)Cluster Abell 370 (z=0.375)

Contours of constant fractional error in the recovered surface density

ParametricParametric –– unknowns: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 – Free-form – unknowns:unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)?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?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?how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use?what prior P(H|I) to use? regularization prior: smoothing linear constraints motivated by knowledge of galaxies, clusters how to estimate errors?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

Known mass distribution:Known mass distribution: N-body cluster

Free-form potential reconstruction with

regularization: Bradac et al. (2005a)

Solve for the potentialSolve for the potential on a grid: 20x20 50x50

Minimize:Minimize:

Error estimationError estimation: bootstrap resampling of weakly lensed galaxies

Rstrongweak 222

Reconstructions:Reconstructions: starting from three input maps; using 210 arclets, 1 four-image system

likelihood moving prior regularization

Cluster RX J1347.5-1145 (z=0.451)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 onemultiple image system to break mass sheet degeneracy

Cluster mass, r<0.5 Mpc =

1.3

M

pc

sunM1510)3.02.1(

ParametricParametric –– unknowns: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 – Free-form – unknowns:unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)?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?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?how many model parameters (# pixels) to use? comparable to # observables; adaptive pixel size greater than # observables what prior P(H|I) to use?what prior P(H|I) to use? regularization prior: source size linear constraints motivated by knowledge of galaxies, clusters how to estimate errors?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

Contours: input mass contoursGray scale: recovered mass

Free-form mass reconstruction with

regularization: Diego et al. (2005b) Known mass distribution:Known mass distribution: 1 large + 3 small NFW profiles

Lens equations:Lens equations: N = [N x M matrix] M N – image positions M – unknowns: mass pixels, source pos.

Pixelate massPixelate mass: start with ~12 x 12 grid, end up with ~500 pixels in a multi-resolution grid.

Sources:Sources: extended, few pixels each

Minimize RMinimize R22: R = N – [N x M] M; residuals vector

Inputs:Inputs:

Prior R2

Initial guess for M unknowns

P(H|I)

P(D|H,I) replaced by linear constraints

Abell 1689, z=0.183106 images from 30 sources[Broadhurst et al. 2005]

Free-form mass reconstruction with

regularization: Diego et al. (2005b) Cluster Abell 1689 (z=0.183)Cluster Abell 1689 (z=0.183)

1 arcmin 185 kpc

contour lines:contour lines: reconstructed mass distribution

map of S/N ratios

Data:Data: 106 images (30 sources) but 601 data pixelsMass pixelsMass pixels: 600, variable size

ErrorsErrors: rms of many reconstructions using different initial conditions (pixel masses, source positions, source redshifts – within error)

ParametricParametric –– unknowns: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 – Free-form – unknowns:unknowns: usually square pixels tiling the lens plane what to solve for (pixelate potential or mass distribution)?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?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?how many model parameters (# pixels) to use? comparable to # observables greater than # observables what prior P(H|I) to use?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?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

Blue – true mass contoursBlack – reconstructed Red – images of point sources

Solve for mass:Solve for mass: ~30x30 grid of mass pixels Data:Data: P(D|H,I) replaced by linear constraints from image pos.Priors P(H|I):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:Ensemble average: 200 models, each reproduces image positions exactly.

Free-form mass reconstruction with

ensemble averaging: PixeLensKnown mass distributionKnown mass distribution

455 images (1 source)

13 images (3 sources)

[Oguri et al. 2004][Inada et al. 2003, 2005][Williams & Saha 2005]

15’’115 kpc

SDSS J1004, zQSO =1.734

blue crosses: galaxies(not used in modeling)red dots: QSO images

... 0.4, 0.2, 0.1, :contours

Fixed constraints:Fixed constraints: positions of 4 QSO images Priors: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 withensemble averaging: PixeLens

Free-form mass reconstruction withensemble averaging: PixeLens

[Oguri et al. 2004][Inada et al. 2003, 2005][Williams & Saha 2005]

15’’115 kpc

SDSS J1004, zQSO =1.734

contours: …-6.25, -3.15, 0, 3.15, 6.25… x 109

MSun/arcsec2dashed 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

Conclusions

PixeLensPixeLens – easy to use, open source lens modeling code, with a GUI interface (Saha & Williams 2004); use to find it.

Galaxy clusters: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: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: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