1. Stochastic Gravitational Lensing and the Nature of Dark Matter Chuck Keeton Rutgers University Gravitational lens database -- http://cfa-www.harvard.edu/castles with: Arthur Congdon (Rutgers), Greg Dobler (Penn), Scott Gaudi (Harvard), Arlie Petters (Duke), Paul Schechter (MIT)

2. 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

3. 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?

4. 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.

5. 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. Clump-hunting: How to find them? cluster of galaxies, ~10 15 M sun single galaxy, ~10 12 M sun (Moore et al. 1999; also Klypin et al. 1999)

6. CDM halos are lumpy Clusters look like this good! cluster of galaxies, ~10 15 M sun single galaxy, ~10 12 M sun (Moore et al. 1999; also Klypin et al. 1999) vs.

7. CDM halos are lumpy cluster of galaxies, ~10 15 M sun single galaxy, ~10 12 M sun (Moore et al. 1999; also Klypin et al. 1999) vs. Galaxies dont bad?

8. 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.

9. 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!

10. S L O Strong Gravitational Lensing Lens equation: The bending is sensitive to all mass, be it luminous or dark, smooth or lumpy.

11. Point Mass Lens Bending angle: Lens equation: Two images for every source position. Source directly behind lens Einstein ring with radius E. 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 84:506)

12. (MACHO project) Microlensing! Data mining: Need to distinguish microlensing from variable stars.

13. Lensing by Galaxies: Hubble Space Telescope Images Double Quad Ring (Zwicky, 1937 Phys Rev 51:290)

14. Radio Lenses 10 = 4+4+2 Double Quad

15. 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.

16. 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:

17. Fermats Principle Time delay surface: Lens equation: Lensed images are critical points of. –minimum –saddle –maximum

18. Lensing and Catastrophe Theory Reinterpet lens equation as a mapping: Jacobian: The critical points of the mapping are important… Observability: image brightness given by

19. Critical curves: det J = 0 (Two curves.) Caustics: Image number changes by 2 Fold and cusp catastrophes. 1 3/2 5/4 Catastrophes in Lensing

20. (Bradac et al. 2002) Substructure complicated catastrophes!

21. (Schechter & Wambsganss 2002)

22. Parametric Mass Modeling Data Positions and brightnesses of the images.3 N img (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

23. 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 : i true i smooth –brightnesses determined by ij : ij true = ij smooth + ij sub

24. 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?

25. 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?

26. 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.

27. folds: A 1 A 2 0 PG 1115+080

28. cusps: A B C 0 B2045+265 (Fassnacht et al. 1999)

29. 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.

30. Connect to observables: R fold vanishes with the distance between the images. But with an unknown coefficient!

32. Derive p(R fold | d 1,d 2 ) A fold depends on: derivatives Physical parameters:galaxy shapes -- from observed galaxy samples tidal shear -- from theoretical models Monte Carlos: Generate ~10 6 mock quads. Extract conditional probability density. What is the range of R fold 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.

33. Archetypal lenses

34. Real lenses

36. 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.

37. Link #2: Theory of Stochastic Lensing Now must understand what happens when we add substructure. Formally, system is described by where i and {p i } are random variables. Images are critical points of random critical point theory. Positions i are independent and identically distributed; and {p i } are independent of i (we hope) marked spatial point process.

38. What I want Given distributions for i and {p i }, I want to compute distributions for the image properties -- especially P( ). 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.

39. Physical Insight Newton: gravity outside a spherical object is insensitive to the objects internal structure.

40. Some analytic results… Implication: To lowest order, all that matters is the average density in substructure.

41. Open questions 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) minimumsaddle

42. Some statistical issues Given p( |{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.

43. 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!

44. OLD SLIDES

45. Optics converging lens diverging lens

46. Gravitational Optics

47. Gravitational Deflection of Light r M Predicted by Einstein, observed by Sir Arthur Eddington in the solar eclipse of 1919.