Stochastic Gravitational Lensing and the Nature of Dark Matter

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Stochastic Gravitational Lensingand 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.harv
ard.edu/castles
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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

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The Preposterous Universe
Can we go beyond merely quantifying dark matter
and dark energy, to learn about fundamental
physics?
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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.

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CDM halos are lumpy
cluster of galaxies,1015 Msun

• Predictions
• Hierarchical structure formationsmall 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.
• Clump-hunting How to find them?

single galaxy,1012 Msun
(Moore et al. 1999 also Klypin et al. 1999)
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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)
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CDM halos are lumpy
cluster of galaxies,1015 Msun
vs.

• Galaxies dont -?bad?

single galaxy,1012 Msun
(Moore et al. 1999 also Klypin et al. 1999)
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A Substructure Crisis?

• CDM seems to overpredict substructure. What does
  it mean?
• Particle physics
• Maybe dark matter isnt 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.

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A Substructure Crisis?

• CDM seems to overpredict substructure. What does
  it mean?
• Particle physics
• Maybe dark matter isnt 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!
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Strong Gravitational Lensing
S
?
?
L
O
Lens equation
The bending is sensitive to all mass, be
itluminous or dark, smooth or lumpy.
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Point Mass Lens

• Bending angle
• Lens equation
• Two images for every source position.
• Source directly behind lens ?Einstein ring with
  radius qE.

sources
lens
2 images of each source
Einstein ring radius
Of course, there is not much hope of observing
this phenomenon directly. (Einstein, 1936
Science 84506)
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Microlensing!
Data mining Need to distinguish microlensing
from variable stars.
(MACHO project)
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Lensing by GalaxiesHubble Space Telescope Images
Double
Quad
Ring
(Zwicky, 1937 Phys Rev 51290)
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Radio Lenses
Quad
10 442
Double
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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.

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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

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Fermats Principle

• Time delay surface
• Lens equation
• Lensed images are critical points of ?.
• minimum
• saddle
• maximum

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Lensing and Catastrophe Theory

• Reinterpet lens equation as a mapping
• Jacobian
• The critical points of the mapping ? are
  important
• Observability image brightness given by

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Catastrophes in Lensing
1
3/2
5/4
Critical curves det J 0 (Two curves.)
Caustics Image number changes by ?2 Fold and
cusp catastrophes.
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Substructure ? complicated catastrophes!
(Bradac et al. 2002)
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(Schechter Wambsganss 2002)
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Parametric Mass Modeling

• Data
• Positions and brightnesses of the images. 3?Nimg
• (Maybe a few other observables.)
• Parameters
• Mass and shape of lens galaxy. 3
• Tidal shear field. 2
• Position and brightness of source. 3
• Substructure. ?

Public software -- http//www.physics.rutgers.edu/
keeton/gravlens
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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 fitrue ? fismooth
• brightnesses determined by fij fijtrue
  fijsmooth fijsub

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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?

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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?

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Link 1 Finding flux ratio anomalies(CRK,
Gaudi Petters 2003 ApJ 598138 2005 ApJ 63535)

• Do the anomalies really indicate
  substructure?Or just a failure of imagination in
  our (parametric) lens models?
• Complaints about model dependenceReal 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.
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folds A1-A2 ? 0
PG 1115080
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cusps A-BC ? 0
B2045265 (Fassnacht et al. 1999)
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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.

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• Connect to observables
• Rfold vanishes with the distance between the
  images.
• But with an unknown coefficient!

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(No Transcript)
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Derive p(Rfold d1,d2)

• Afold depends on
• ?f 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.

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Archetypal lenses
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Real lenses
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Real lenses
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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)
• Catastrophe theory reveals generic features
  which guide data analysis
• and provide a rigorous foundation for
  substructure studies.

Substructure exists, and is relatively common.
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Link 2 Theory of Stochastic Lensing

• Now must understand what happens when we add
  substructure.
• Formally, system is described by
• where qi and pi are random variables.
• Images are critical points of ??? random critical
  point theory.
• Positions qi are independent and identically
  distributed and pi are independent of qi (we
  hope) ? marked spatial point process.

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What I want

• Given distributions for qi 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 cant do the (inverse) Fourier transforms.

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Physical Insight

• Newton gravity outside a spherical object is
  insensitive to the objects internal structure.
• ??????????????????????????????????????????????????
  ???

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Some analytic results
Implication To lowest order, all that matters is
the average density in substructure.
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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)
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Some statistical issues

• Given p(msub), 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.

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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!

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OLD SLIDES
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Optics
converging lens
diverging lens
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Gravitational Optics
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Gravitational Deflection of Light
r
M
Predicted by Einstein, observed by SirArthur
Eddington in the solar eclipse of 1919.