Stochastic Gravitational Lensing
and the Nature of Dark Matter
Chuck Keeton
Rutgers University
with:
Arthur Congdon (Rutgers), Greg Dobler (Penn),
Scott Gaudi (Harvard), Arlie Petters (Duke),
Paul Schechter (MIT)
Gravitational lens database -- http://cfa-www.harvard.edu/castles
Outline
• Cold Dark Matter 101
• Gravitational Lensing 101/201
• Evidence for dark matter substructure
– catastrophe theory
• Stochastic gravitational lensing
– random critical point theory
– marked spatial point processes
• Some statistical issues
– Bayesian inference
– small datasets
– testing relations, not just parameters
The Preposterous Universe
4%
baryons: stars and gas
(all we can ever see)
23%
dark matter: non-baryonic; exotic
73%
dark energy: cosmic repulsion; perhaps
vaccuum energy or quintessence
Can we go beyond merely quantifying
dark matter and dark energy, to learn
about fundamental physics?
The Cold Dark Matter (CDM) Paradigm
• Dark matter is assumed to be
– “cold”: non-relativistic
– “collisionless”: only feels gravity
– axions, neutralinos, lightest supersymmetric particle, …
• Successful in explaining large-scale properties of the universe.
– global geometry, distribution of galaxies, cosmic microwave background, …
• Successful in describing many features of galaxies and clusters.
– the “missing mass”
• But several challenges (crises?) related to the distribution of dark matter
on small scales.
cluster of galaxies,
~1015 Msun
CDM halos are lumpy
Predictions:
• Hierarchical structure formation:
small objects form first, then aggregate
into larger objects.
• Small objects are dense, so they can
maintain their integrity during mergers.
• Large halos contain the remnants of
their many progenitors - substructure.
single galaxy,
~1012 Msun
(Moore et al. 1999; also Klypin et al. 1999)
• Clump-hunting: How to find them?
CDM halos are lumpy
cluster of galaxies,
~1015 Msun
vs.
Clusters look like this - good!
single galaxy,
~1012 Msun
(Moore et al. 1999; also Klypin et al. 1999)
CDM halos are lumpy
cluster of galaxies,
~1015 Msun
vs.
Galaxies don’t - bad?
single galaxy,
~1012 Msun
(Moore et al. 1999; also Klypin et al. 1999)
A Substructure Crisis?
CDM seems to overpredict substructure. What does it mean?
Particle physics
• Maybe dark matter isn’t cold and collisionless. (CDM is wrong!)
• Maybe it is warm, self-interacting, fuzzy, sticky, …
Astrophysics
• We only see clumps if they contain stars and/or gas.
• Maybe astrophysical processes suppress star formation in small objects,
so most clumps are invisible.
A Substructure Crisis?
CDM seems to overpredict substructure. What does it mean?
Particle physics
• Maybe dark matter isn’t cold and collisionless. (CDM is wrong!)
• Maybe it is warm, self-interacting, fuzzy, sticky, …
Astrophysics
• We only see clumps if they contain stars and/or gas.
• Maybe astrophysical processes suppress star formation in small objects,
so most clumps are invisible.
Need to search for a large population
of “invisible” objects!
Strong Gravitational Lensing
S

L

Lens equation:
The bending is sensitive to all mass, be it
luminous or dark, smooth or lumpy.
O
Point Mass Lens
• Bending angle:
sources
• Lens equation:
lens
2 images of
each source
• Two images for every source position.
• Source directly behind lens 
Einstein ring with radius E.
Einstein
ring radius
“Of course, there is not much hope of observing this phenomenon directly.”
(Einstein, 1936 Science 84:506)
Microlensing!
Data mining: Need to distinguish
microlensing from variable stars.
(MACHO project)
“Double”
Lensing by Galaxies:
Hubble Space Telescope Images
“Quad”
“Ring”
(Zwicky, 1937 Phys Rev 51:290)
Quad
Radio Lenses
10 = 4+4+2
Double
What is lensing good for?
Strong lensing
• Multiple imaging of some distant source.
• Used to study the dark matter halos of galaxies and clusters of galaxies.
Microlensing
• Temporary brightening of a star in our galaxy.
• Used to probe for dark stellar-mass objects in our own galaxy.
Weak lensing
• Small, correlated distortions in the shapes of distant galaxies.
• Used to study the large-scale distribution of matter in the universe.
Extended Mass Distributions: 2-d Gravity
• Work with 2-d angle vectors on the sky.
• Interpret bending angle as 2-d gravity force  gradient of 2-d
gravitational potential.
• Extended mass distribution:
• General lens equation:
Fermat’s Principle
• Time delay surface:
• Lens equation:
• Lensed images are critical points of .
– minimum
– saddle
– maximum
Lensing and Catastrophe Theory
• Reinterpet lens equation as a mapping:
• Jacobian:
• The critical points of the mapping  are important…
• Observability: image brightness given by
Catastrophes in Lensing
1
3/2
5/4
Critical curves:
det J = 0
Caustics:
Image number
changes by 2
(Two curves.)
Fold and cusp catastrophes.
Substructure  complicated catastrophes!
(Bradac et al. 2002)
(Schechter & Wambsganss 2002)
Parametric Mass Modeling
Data
• Positions and brightnesses of the images.
• (Maybe a few other observables.)
Parameters
• Mass and shape of lens galaxy.
• Tidal shear field.
• Position and brightness of source.
• Substructure.
3Nimg
…
3
2
3
?
Public software -- http://www.physics.rutgers.edu/~keeton/gravlens
Lensing and Substructure
Fact
• In 4-image lenses, the image positions can be fit by smooth lens models.
• The flux ratios cannot.
Interpretation
• Flux ratios are perturbed by substructure in the lens potential.
(Mao & Schneider 1998;
Metcalf & Madau 2001;
Dalal & Kochanek 2002)
• Recall:
– positions determined by i:
– brightnesses determined by ij:
itrue  ismooth
ijtrue = ijsmooth + ijsub
Substructure Statistics
• Can always(?) add one or two clumps and get a good model.
• More interesting are clump population statistics. Are they:
– Consistent with known populations of substructure?
(globular clusters, dwarf galaxies, …)
– Consistent with CDM predictions?
– None of the above?
From Lensing to Dark Matter Physics
• Find lenses with flux ratio anomalies.
– catastrophe theory
• How do the statistics of anomalies depend on properties of the substructure
population?
– random critical point theory
– marked spatial point processes
• Measure properties of substructure population.
– Bayesian inference
– small datasets
• Compare with CDM predictions.
– testing relations, not just parameters
• How do substructure population statistics depend on physical properties of
dark matter?
Link #1: Finding flux ratio anomalies
(CRK, Gaudi & Petters 2003 ApJ 598:138; 2005 ApJ 635:35)
• Do the “anomalies” really indicate substructure?
Or just a failure of imagination in our (parametric) lens models?
• Complaints about “model dependence”…
Real problem is use of global failures to probe local features.
• Fortunately, catastrophe theory enables a local lensing analysis
that leads to some generic statements…
Use mathematical theory to develop a statistical analysis
to apply to astronomical data.
folds: A1-A2  0
PG 1115+080
cusps: A-B+C  0
B2045+265 (Fassnacht et al. 1999)
Theory of fold catastrophes in lensing
• Jacobian:
• Fold critical point: (in appropriate coordinates)
• General perturbation theory analysis near fold point:
At lowest order, the two images mirror one another.
• Connect to observables:
• Rfold vanishes with the distance between the images.
• But with an unknown coefficient!
Derive p(Rfold | d1,d2)
Afold depends on:
•
 derivatives
• Physical parameters:
galaxy shapes -- from observed galaxy samples
tidal shear -- from theoretical models
Monte Carlos:
• Generate ~106 mock quads.
• Extract conditional probability density.
What is the range of Rfold in realistic smooth lenses?
If real lenses lie outside this range, they must not be smooth.

substructure.
Analysis relies on generic properties of fold catastrophes.
Archetypal lenses
Real lenses
Real lenses
The Fold and Cusp Relations
Violations of the generic relations:
• 5 anomalies among 12 fold lenses
• 3 anomalies among 4 cusp lenses
• (No firm conclusions about 6 cross lenses)
Substructure exists, and is
relatively common.
Catastrophe theory reveals generic features … which guide data analysis
… and provide a rigorous foundation for substructure studies.
Link #2: Theory of Stochastic Lensing
• Now must understand what happens when we add substructure.
• Formally, system is described by
where i and {pi} are random variables.
• Images are critical points of  random critical point theory.
• Positions i are independent and identically distributed; and {pi} are
independent of i (we hope)  marked spatial point process.
What I want
• Given distributions for i and {pi}, I want to compute distributions for
the image properties -- especially P(m).
• Analytically, if possible.
– Explore large parameter spaces.
– Gain general insights, not just specific results.
• Clumps are independent and identically distributed  could use
characteristic function method.
• But I can’t do the (inverse) Fourier transforms.
Physical Insight
Newton: gravity outside a spherical object is insensitive to the object’s
internal structure.

Some analytic results…
Implication: To lowest order, all that matters
is the average density in substructure.
Open questions
minimum
saddle
• For certain kinds of substructure,
minima and saddles respond in
opposite directions.
• But which direction?
• Why?
• How generic is that result?
• Signal seems to be present in
data; what does it tell us about
substructure?
(Schechter & Wambsganss 2002)
Some statistical issues
• Given p(m|{sub}), use Bayesian inference to constrain substructure
parameters.
• Current data: 22 quad lenses
– 8 anomalies in 16 fold/cusp lenses
– ? anomalies in 6 cross lenses
• Future samples: 100s or 1000s, each with its own probability density.
• To test dark matter physics, will want to examine relations.
Conclusions
• Gravitational lensing is a unique probe of dark matter.
• Flux ratio anomalies  substructure  dark matter physics.
• Can do brute force analysis. But interdisciplinary approach yields much
deeper results.
– We can reliably identify anomalies.
– We can understand what aspects of substructure we can measure.
– We will eventually understand how substructure probes dark matter physics.
• We pose interesting math/stats questions … then use the answers to do
exciting physics/astronomy!
OLD SLIDES
Optics
converging lens
diverging lens
Gravitational Optics
Gravitational Deflection of Light
r
M
Predicted by Einstein, observed by Sir
Arthur Eddington in the solar eclipse of 1919.