Outline
Galaxy cluster Abell 1689
 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
A Brief Introduction to Lensing
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
total travel time
position on the sky
Goal: find positions of images on the plane of the sky
A Brief Introduction to Lensing
Plane of the sky
Elliptical lens
Off-axis source
Circularly symmetric lens
Off-axis source
Circularly symmetric lens
On-axis source
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]
Substructure and Image Properties
Maxima, minima, saddles of the arrival time surface correspond to images
smooth elliptical lens
… with mass lump (~1%) added
Examples of Lens Systems
Galaxies
~1 arcsecond
Galaxy Clusters
~ 1 arcminute
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
prior
likelihood
P( H | I )  P( D | H , I )
P ( H | D, I ) 
P( D | I )
evidence
#data > #model parameters
P(D|H,I) dominates
P(H|I) not important
#data < #model parameters
P(H|I) is important !
parametric methods
5-10 parameters
P(H|I) choices:
•maximum entropy
•min. w.r.t. observed light
•smoothing (local, global)
•…
D is data with errors
P(D|H,I) is the usual c2-type fcn
P(H|I) provides regularization
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
PixeLens
posterior
Mass Modeling Methods
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: 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 c2-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)
Parametric mass reconstruction:
Kneib et al. (1996), Natarajan et al. (2002)
Question: what is the size of cluster galaxies?
Each galaxy’s mass, radius are fcn (Lum)
galaxy + cluster mass are superimposed
Maximize P(D|H,I) likelihood fcn
Abell 2218,
z=0.175
collisionless
DM predictions
collisional
fluid-like DM
predictions
520 kpc
Within 1 Mpc of cluster center
galaxies comprise 10-20% of mass;
consistent with collisionless DM
Mass Modeling Methods
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: 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 c2-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)
Free-form mass reconstruction with
regularization: AbdelSalam et al. (1998)
Lens eqn is linear in the unknowns: mass pixels, source positions



 src  im   pix  pix f (im )
Image elongations also provide linear constraints.
Data: coords, elongations of 9 images (4 sources) & 18 arclets
Pixelate mass distribution ~ 3000 pixels (unknowns)
Regularize w.r.t. light distribution
Errors: rms of mass maps with randomized light distribution
260 kpc
Cluster Abell 2218
(z=0.175)
P(D|H,I)
replaced by
linear
constraints
P(H|I)
Free-form mass reconstruction with
regularization: AbdelSalam et al. (1998)
Cluster Abell 2218
(z=0.175)
Overall,
mass distribution
follows light, but:
center of mass
center of light
Mass/Light ratios are displaced
by ~ 30 kpc
of 3 galaxies
(~ 3 x Sun’s dist.
differ by x 10
Chandra X-ray emission elongated “horizontally”;
X-ray peak close to the predicted mass peak.
Machacek et al. (2002)
from Milky Way’s
center)
Free-form mass reconstruction with
regularization: AbdelSalam et al. (1997)
Cluster Abell 370
(z=0.375)
Color map: optical image of the cluster
Contours: recovered surface density map
Regularized 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)
Contours of constant fractional error
in the recovered surface density
Mass Modeling Methods
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: 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 c2-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)
Free-form potential reconstruction with
regularization: Bradac et al. (2005a)
Known mass distribution: N-body cluster
Solve for the potential on a grid: 20x20  50x50
2
2
2
c

c

c
weak
strong  R
Minimize:
likelihood
moving prior
regularization
Error estimation: bootstrap resampling of weakly
lensed galaxies
Reconstructions: starting from three input maps;
using 210 arclets, 1 four-image system
Free-form potential reconstruction with
regularization: Bradac et al. (2005b)
1.3 Mpc
Cluster RX J1347.5-1145
(z=0.451)
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 =
(1.2  0.3) 1015 M sun
Mass Modeling Methods
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: 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 c2-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)
Free-form mass reconstruction with
regularization: Diego et al. (2005b)
Known mass distribution:
1 large + 3 small NFW profiles
Lens equations:
P(D|H,I) replaced
N = [N x M matrix] M by linear constraints
N – image positions
M – unknowns: mass pixels, source pos.
Pixelate mass:
start with ~12 x 12 grid,
end up with ~500 pixels in
a multi-resolution grid.
Sources:
extended, few pixels each
Minimize R2:
R = N – [N x M] M; residuals vector
Contours: input mass contours
Gray scale: recovered mass
Inputs:
 P(H|I)
Prior R2
Initial guess for M unknowns
Abell 1689,
z=0.183
106 images from 30 sources
[Broadhurst et al. 2005]
Free-form mass reconstruction with
regularization: Diego et al. (2005b)
Cluster Abell 1689 (z=0.183)
Data: 106 images (30 sources)
but 601 data pixels
Mass pixels: 600, variable size
1 arcmin
185 kpc
contour lines:
reconstructed mass distribution
Errors: rms of many reconstructions
using different initial conditions
(pixel masses, source positions,
source redshifts – within error)
map of S/N ratios
Mass Modeling Methods
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: 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 c2-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)
Free-form mass reconstruction with
ensemble averaging: PixeLens
Known mass distribution
Solve for mass:
~30x30 grid of mass pixels
Data:
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  45 of radial
 -0.1 < 2D density slope < -3
 (no smoothness constraint)
Ensemble average:
200 models, each reproduces
image positions exactly.
Blue – true mass contours
Black – reconstructed
Red – images of point sources
5 images (1 source)
13 images (3 sources)
Free-form mass reconstruction with
ensemble averaging: PixeLens
SDSS J1004,
zQSO =1.734
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
15’’
115 kpc
blue crosses: galaxies
(not used in modeling)
red dots: QSO images
[Oguri et al. 2004]
[Inada et al. 2003, 2005]
[Williams & Saha 2005]
contours :   0.1, 0.2, 0.4, ...
Free-form mass reconstruction with
ensemble averaging: PixeLens
SDSS J1004,
zQSO =1.734
19 galaxies within 120 kpc of cluster center:
comprise <10% of mass, have 3<Mass/Light<15
 galaxies were stripped of their DM
Mass maps of residuals for 2 PixeLens reconstructions
15’’
115 kpc
blue crosses: galaxies
(not used in modeling)
red dots: QSO images
[Oguri et al. 2004]
[Inada et al. 2003, 2005]
[Williams & Saha 2005]
density slope -1.25
density slope -0.39
contours: …-6.25, -3.15, 0, 3.15, 6.25…
dashed
solid
x
109 MSun/arcsec2
Conclusions
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
PixeLens – easy to use, open source lens modeling code, with a GUI
interface (Saha & Williams 2004); use
to find it.