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Probing Small-Scale Structure in Galaxies with Strong Gravitational Lensing
Arthur Congdon • Rutgers University
Strong Gravitational Lensing
~
S
L O
Lensing is sensitive to all mass, be itluminous or dark, smooth or lumpy
quasar,
z ~ 15
galaxy, z ~ 0.21
Lens equation:
~
OS
LS
D
Dwhere
Lensing by a Singular Isothermal Sphere (SIS)
• Reduced deflection angle:
• Lens equation:
• Source directly behind lens produces Einstein ring with angular “radius” E
EOS
LS
D
D
c
2
2
E
SIS Lensing
Courtesy of S. Rappaport
SIS with Shear
• Lens equation:
xyx
xxu E
22
yyx
yyv E
22
• 1, 2 or 4 images can be produced
SIS with Shear
Courtesy of S. Rappaport
CASTLES ~ www.cfa.harvard.edu/castles
Lensing by Galaxies:Hubble Space Telescope
Images
“Double”
“Quad”
“Ring”
Quasars as Lensed Sources
• Radio emission comes from extended jets
• Optical, UV and X-ray emission comes mainly from the central accretion disk
Via Lactea CDM simulation(Diemand et al. 2007)
Via Lactea CDM simulation(Diemand et al. 2007)
• Hierarchical structure formation:small objects form first, then aggregate into larger objects
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
Clusters look like this good!
cluster of galaxies,~1015 Msun
single galaxy,~1012 Msun
(Moore et al. 1999)
vs.
cluster of galaxies,~1015 Msun
single galaxy,~1012 Msun
vs.
Galaxies don’t - bad?
(Moore et al. 1999)
Strigari et al (2007)
Missing Satellites Problem
Multipole Models of Four-Image Gravitational Lenses with Anomalous Flux Ratios
MNRAS 364:1459 (2005)
Four-Image Lenses
Source plane
Image plane
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
• Deviations small-scale structure
1foldfold dAFF
FFR
BA
BA
BA
BA
21cuspcusp dA
FFF
FFFR
CBA
CBA
CBA
CBA
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?
07.0006.0 sub f
“Minimum Wiggle” Model
• Allow many multipoles, up to mode kmax
• Models underconstrained large solution space
• Minimize departures from elliptical symmetry.
B2045+265
Multipole Formalism
• Convergence:
• Lens potential:
• 2-D Poisson equation:
• Fourier expansion:
)(2
1),( G
rr
)(),( rFr
22 )()()( '' FFG
)]sin()cos([2
)(max
1
0 kbkaa
F k
k
kk
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:
)X(XU
X
Udet1
bχA�
•Use SVD to solve for parameters when kmax>4:
•Minimize departure from elliptical symmetry (i.e., minimize wiggles):
•Adding shear leads to nonlinear equations
i
iic )()0(
max
3
22222
2 18
1 k
kkk bakr
Solving for Unknowns
Solution for B2045+265
9max k 17max k
Solution for B2045+265
Isodensity contours (solid) and critical curves (dashed)
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
Analytic Relations for Magnifications and Time Delays
in Gravitational Lenses withFold and Cusp Configurations
Submitted to J. Math. Phys.
Lens Time Delays
Q0957+561
Kundić et al. (1997)
Robust probe of dark matter substructure?
Local Coordinates
Caustic (source plane) Critical curve (image plane)
fold cusp
Perturbation Theory for Fold Lenses
• Lens Potential:
42
321
22
212
31
41
32
2212
21
31
22
2121 2
1)1(
2
1),(
rpnmk
hgfeK
• Lens Equation:
• For small displacements: (u1, u2) (u1, u2)
111
u2
22
u
• Expand image positions in ε:
• Solve for coefficients to find:
• Image separation:
2/32
2/122
2/31
2/111
)(
)(
O
O
2/32
22
12/122
2/3211
)(9
3
3
)(3
3
OKh
ugghu
h
u
OhK
guhu
2/32/121 )(
3
4 Oh
ud
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
212
222
11 ,2
1 uu
31
22/32 2
)()(27
16d
hOu
hfold
Comparison to “Exact” Numerical Solution
Analytic scaling is astrophysically relevant
E E
Using Differential Time Delays to Identify Gravitational Lenses with
Small-Scale Structure
In preparation for submission to ApJ
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
Variation of halong Caustic
Variation of halong Caustic
Variation of halong Caustic
Time Delays for aRealistic 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
Matching Mock and Observed Lenses
Histograms for Scaled Time Delay: Folds
PG 1115+080
SDSS J1004+4112
Histograms for Scaled Time Delay: Cusps
RX J0911+0551
RX J1131-1231
Histograms for Time Delay Ratios: Folds
B1608+656
HE 0230-2130
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
Acknowledgments
I would like to thank my collaborators,
Chuck Keeton and Erik Nordgren