1. Probing Small-Scale Structure in Galaxies with Strong Gravitational Lensing Arthur Congdon Rutgers University

3. Strong Gravitational Lensing S L O  Lensing is sensitive to all mass, be it luminous or dark, smooth or lumpy quasar, z ~ 1  5 galaxy, z ~ 0.2  1 Lens equation:where

4. Lensing by a Singular Isothermal Sphere (SIS) Reduced deflection angle: Lens equation: Source directly behind lens produces Einstein ring with angular “radius”  E

5. SIS Lensing Courtesy of S. Rappaport

6. SIS with Shear Lens equation: 1, 2 or 4 images can be produced

7. SIS with Shear Courtesy of S. Rappaport

8. CASTLES ~ www.cfa.harvard.edu/castles Lensing by Galaxies: Hubble Space Telescope Images “Double” “Quad” “Ring”

9. Quasars as Lensed Sources Radio emission comes from extended jets Optical, UV and X-ray emission comes mainly from the central accretion disk

10. Via Lactea CDM simulation (Diemand et al. 2007)

11. Via Lactea CDM simulation (Diemand et al. 2007) Hierarchical structure formation: small objects form first, then aggregate into larger objects

12. Via Lactea CDM simulation (Diemand et al. 2007) Hierarchical structure formation: small objects form first, then aggregate into larger objects Large halos contain the remnants of their many progenitors  substructure

13. Clusters look like this  good! cluster of galaxies, ~10 15 M sun single galaxy, ~10 12 M sun (Moore et al. 1999) vs.

14. cluster of galaxies, ~10 15 M sun single galaxy, ~10 12 M sun vs. Galaxies don’t - bad? (Moore et al. 1999)

15. Strigari et al (2007) Missing Satellites Problem

16. Multipole Models of Four-Image Gravitational Lenses with Anomalous Flux Ratios MNRAS 364:1459 (2005)

17. Four-Image Lenses Source plane Image plane

18. Universal Relations for Folds and Cusps Flux relation for a fold pair (Keeton et al. 2005): Flux relation for a cusp triplet (Keeton et al. 2003): Valid for all smooth mass models D eviations small-scale structure

19. Flux Ratio Anomalies Many lenses require small-scale structure (Mao & Schneider 1998; Keeton, Gaudi & Petters 2003, 2005) Could be CDM substructure (Metcalf & Madau 2001; Chiba 2002) Fitting the lenses requires (Dalal & Kochanek 2002) Broadly consistent with CDM Is substructure the only viable explanation?

20. “Minimum Wiggle” Model Allow many multipoles, up to mode k max Models underconstrained  large solution space Minimize departures from elliptical symmetry. B2045+265

21. Multipole Formalism Convergence: Lens potential: 2-D Poisson equation: Fourier expansion:

22. Observational Constraints Lens equation: Image positions give 2n constraints Magnification: Flux ratios give n-1 constraints Combine 3n-1 constraints into a single matrix equation:

23. Use SVD to solve for parameters when k max >4: Minimize departure from elliptical symmetry (i.e., minimize wiggles): Adding shear leads to nonlinear equations Solving for Unknowns

24. Solution for B2045+265

25. Isodensity contours (solid) and critical curves (dashed)

26. What Have We Learned from Multipoles? Multipole models with shear cannot explain anomalous flux ratios Isodensity contours remain wiggly, regardless of truncation order Wiggles are most prominent near image positions; implies small-scale structure Ruled out a broad class of alternatives to CDM substructure

27. Analytic Relations for Magnifications and Time Delays in Gravitational Lenses with Fold and Cusp Configurations Submitted to J. Math. Phys.

28. Lens Time Delays Q0957+561 Kundić et al. (1997) Robust probe of dark matter substructure?

29. Local Coordinates Caustic (source plane)Critical curve (image plane) foldcusp

30. Perturbation Theory for Fold Lenses Lens Potential: Lens Equation: For small displacements: (u 1, u 2 ) (  u 1,  u 2 )

31. Expand image positions in ε: Solve for coefficients to find: Image separation:

32. Time-Delay Relation for Fold Pairs Scaled Time Delay: Use perturbation theory to get differential time delay: Time-delay anomalies may provide a more sensitive probe of small-scale structure than flux-ratio anomalies

33. Comparison to “Exact” Numerical Solution Analytic scaling is astrophysically relevant  E 

34. Using Differential Time Delays to Identify Gravitational Lenses with Small-Scale Structure In preparation for submission to ApJ

35. Dependence of Time Delay on Lens Potential and Position along Caustic Use h as proxy for time delay Model lens galaxy as SIE with shear Higher-order multipoles are not so important here

36. Variation of h along Caustic

39. Time Delays for a Realistic Lens Population Perform Monte Carlo simulations: –use galaxies with distribution of ellipticity, octopole moment and shear –use random source positions to create mock four- image lenses –use Gravlens software (Keeton 2001) to obtain image positions and time delays –create time delay histogram for each image pair

40. Matching Mock and Observed Lenses

41. Histograms for Scaled Time Delay: Folds PG 1115+080 SDSS J1004+4112

42. Histograms for Scaled Time Delay: Cusps RX J0911+0551 RX J1131-1231

43. Histograms for Time Delay Ratios: Folds B1608+656 HE 0230-2130

44. What Have We Learned from Time Delay Analytics and Numerics? Time delay of the close pair in a fold lens scales with the cube of image separation Time delay is sensitive to ellipticity and shear, but not higher-order multipoles For a given image separation and lens potential, the time delay remains constant if the source is not near a cusp Monte Carlo simulations reveal strong time-delay anomalies in RX J0911+0551 and RX J1131-1231

45. Acknowledgments I would like to thank my collaborators, Chuck Keeton and Erik Nordgren