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Probing Small-Scale Structure in Galaxies with Strong Gravitational Lensing
Arthur Congdon • Rutgers University
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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
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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
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SIS Lensing
Courtesy of S. Rappaport
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SIS with Shear
• Lens equation:
xyx
xxu E
22
yyx
yyv E
22
• 1, 2 or 4 images can be produced
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SIS with Shear
Courtesy of S. Rappaport
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CASTLES ~ www.cfa.harvard.edu/castles
Lensing by Galaxies:Hubble Space Telescope
Images
“Double”
“Quad”
“Ring”
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Quasars as Lensed Sources
• Radio emission comes from extended jets
• Optical, UV and X-ray emission comes mainly from the central accretion disk
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Via Lactea CDM simulation(Diemand et al. 2007)
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Via Lactea CDM simulation(Diemand et al. 2007)
• Hierarchical structure formation:small objects form first, then aggregate into larger objects
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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
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Clusters look like this good!
cluster of galaxies,~1015 Msun
single galaxy,~1012 Msun
(Moore et al. 1999)
vs.
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cluster of galaxies,~1015 Msun
single galaxy,~1012 Msun
vs.
Galaxies don’t - bad?
(Moore et al. 1999)
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Strigari et al (2007)
Missing Satellites Problem
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Multipole Models of Four-Image Gravitational Lenses with Anomalous Flux Ratios
MNRAS 364:1459 (2005)
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Four-Image Lenses
Source plane
Image plane
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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
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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
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“Minimum Wiggle” Model
• Allow many multipoles, up to mode kmax
• Models underconstrained large solution space
• Minimize departures from elliptical symmetry.
B2045+265
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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
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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�
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•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
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Solution for B2045+265
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9max k 17max k
Solution for B2045+265
Isodensity contours (solid) and critical curves (dashed)
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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
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Analytic Relations for Magnifications and Time Delays
in Gravitational Lenses withFold and Cusp Configurations
Submitted to J. Math. Phys.
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Lens Time Delays
Q0957+561
Kundić et al. (1997)
Robust probe of dark matter substructure?
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Local Coordinates
Caustic (source plane) Critical curve (image plane)
fold cusp
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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
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• 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
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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
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Comparison to “Exact” Numerical Solution
Analytic scaling is astrophysically relevant
E E
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Using Differential Time Delays to Identify Gravitational Lenses with
Small-Scale Structure
In preparation for submission to ApJ
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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
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Variation of halong Caustic
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Variation of halong Caustic
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Variation of halong Caustic
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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
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Matching Mock and Observed Lenses
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Histograms for Scaled Time Delay: Folds
PG 1115+080
SDSS J1004+4112
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Histograms for Scaled Time Delay: Cusps
RX J0911+0551
RX J1131-1231
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Histograms for Time Delay Ratios: Folds
B1608+656
HE 0230-2130
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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
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Acknowledgments
I would like to thank my collaborators,
Chuck Keeton and Erik Nordgren