Cosmological Applications of Gravitational lensing

1
Gravitational Lensing Mass Reconstruction Methods
and Results
Liliya L.R. Williams (U Minnesota) Prasenjit Saha
(QMW, London Univ. of Zurich)
2
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

3
A Brief Introduction to Lensing
Goal find positions of images on the plane of
the sky 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
position on the sky
total travel time
4
A Brief Introduction to Lensing
Plane of the sky
Circularly symmetric lens On-axis source
Circularly symmetric lens Off-axis source
Elliptical lens Off-axis source
5
All the Information about Imagesis 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
Maxima, minima, saddles of the arrival time
surface correspond to images
smooth elliptical lens
with mass lump (1) added
7
Examples of Lens Systems
Galaxy Clusters
Galaxies
1 arcminute
1 arcsecond

• 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
prior likelihood
posterior

• P(HI) choices
• maximum entropy
• min. w.r.t. observed light
• smoothing (local, global)

evidence
parametric methods 5-10 parameters
data gt model parameters P(DH,I)
dominates P(HI) not important dat
a lt model parameters P(HI) is
important !
D is data with errors P(DH,I) is the usual
c2-type fcn P(HI) provides regularization
D is exact (perfect data) P(DH,I) is replaced
by linear constraints P(HI) can use additional
constraints P(HI) can also provide regularization
D is exact (perfect data) P(DH,I) is replaced by
linear constraints P(HI) is replaced by linear
constraints no regularization -gt ensemble average
PixeLens
9
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(DH,I) is usually a c2-type function
data without errors P(DH,I) replaced
by linear constraints how many model
parameters ( pixels) to use?
comparable to observables
greater than observables what
prior P(HI) 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)
10
Parametric mass reconstructionKneib et al.
(1996), Natarajan et al. (2002)
Question what is the size of cluster
galaxies? Each galaxys mass, radius are fcn
(Lum) galaxy cluster mass are superimposed
Maximize P(DH,I) likelihood fcn
Abell 2218, z0.175
collisionless DM predictions
Best fit to 25 galaxies
collisional fluid-like DM predictions
Within 1 Mpc of cluster center galaxies comprise
10-20 of mass consistent with collisionless DM
520 kpc
11
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(DH,I) is usually a c2-type function
data without errors P(DH,I) replaced
by linear constraints how many model
parameters ( pixels) to use?
comparable to observables
greater than observables what
prior P(HI) 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)
12
Free-form mass reconstruction withregularization
AbdelSalam et al. (1998)
Lens eqn is linear in the unknowns mass pixels,
source positions




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
P(DH,I) replaced by linear constraints
P(HI)
Cluster Abell 2218 (z0.175)
260 kpc
13
Free-form mass reconstruction withregularization
AbdelSalam et al. (1998)
Cluster Abell 2218 (z0.175)
center of mass center of light are
displaced by 30 kpc ( 3 x Suns dist. from
Milky Ways center)
Overall, mass distribution follows light, but
Mass/Light ratios of 3 galaxies differ by x 10
Chandra X-ray emission elongated
horizontally X-ray peak close to
the predicted mass peak.
centroid
peak
Machacek et al. (2002)
14
Free-form mass reconstruction withregularization
AbdelSalam et al. (1997)
Cluster Abell 370 (z0.375)

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
15
Free-form mass reconstruction withregularization
AbdelSalam et al. (1997)
Cluster Abell 370 (z0.375)
Contours of constant fractional error in the
recovered surface density
16
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(DH,I) is usually a c2-type function
data without errors (perfect data)
P(DH,I) replaced by linear constraints
how many model parameters ( pixels) to use?
comparable to observables
greater than observables
what prior P(HI) 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)
17
Free-form potential reconstruction
withregularization Bradac et al. (2005a)
Known mass distribution N-body cluster
Solve for the potential on a grid 20x20 ? 50x50
Minimize Error estimation bootstrap
resampling of weakly
lensed galaxies
likelihood moving prior
regularization
Reconstructions starting from three input
maps using 210
arclets, 1 four-image system
18
Free-form potential reconstruction
withregularization Bradac et al. (2005b)
Cluster RX J1347.5-1145 (z0.451)
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, rlt0.5 Mpc
1.3 Mpc
19
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(DH,I) is usually a c2-type function
data without errors (perfect data)
P(DH,I) replaced by linear constraints
how many model parameters ( pixels) to use?
comparable to observables
adaptive pixel size greater
than observables what prior P(HI)
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)
20
Free-form mass reconstruction withregularization
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 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 Inputs Prior R2
Initial guess for M unknowns
P(DH,I) replaced by linear constraints
Contours input mass contours Gray scale
recovered mass
? P(HI)
21
Abell 1689, z0.183 106 images from 30
sources Broadhurst et al. 2005
22
Free-form mass reconstruction withregularization
Diego et al. (2005b)
Cluster Abell 1689 (z0.183)
Errors rms of many reconstructions
using different initial conditions
(pixel masses, source positions,
source redshifts within error)
Data 106 images (30 sources) but 601
data pixels Mass pixels 600, variable size
map of S/N ratios
1 arcmin 185 kpc
contour lines reconstructed mass distribution
23
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(DH,I) is usually a c2-type function
data without errors (perfect data)
P(DH,I) replaced by linear constraints
how many model parameters ( pixels) to use?
comparable to observables
greater than observables
what prior P(HI) 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)
24
Free-form mass reconstruction withensemble
averaging PixeLens
Known mass distribution

• Solve for mass
• 30x30 grid of mass pixels
• Data
• P(DH,I) replaced by linear
• constraints from image pos.
• Priors P(HI)
• mass pixels non-negative
• lens center known
• density gradient must point
• within of radial
• -0.1 lt 2D density slope lt -3
• (no smoothness constraint)
• Ensemble average
• 200 models, each reproduces
• image positions exactly.

5 images (1 source)
Blue true mass contours Black reconstructed
Red images of point sources
13 images (3 sources)
25
Free-form mass reconstruction withensemble
averaging PixeLens

• Fixed constraints
  positions of 4 QSO images
• Priors
• external shear PA 10 45 deg. (Oguri et al.
  2004)
• -0.25 lt 2D density slope lt -3.0
• density gradient direction constraint
• must point within 45 or 8 deg. from radial

SDSS J1004, zQSO 1.734
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
26
Free-form mass reconstruction withensemble
averaging PixeLens
SDSS J1004, zQSO 1.734
19 galaxies within 120 kpc of cluster
center comprise lt10 of mass, have
3ltMass/Lightlt15 ? 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
density slope -1.25
density slope -0.39
Oguri et al. 2004 Inada et al. 2003,
2005 Williams Saha 2005
contours -6.25, -3.15, 0, 3.15, 6.25 x 109
MSun/arcsec2
dashed solid
27
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.