Probing Small-Scale Structure in Galaxies
with Strong Gravitational Lensing
Arthur Congdon • Rutgers University
Strong Gravitational Lensing
~
S

quasar,
z ~ 15

L
galaxy, z ~ 0.21
Lens equation:
   
where
DLS ~


DOS
Lensing is sensitive to all mass, be it
luminous or dark, smooth or lumpy
O
Lensing by a Singular Isothermal Sphere (SIS)
• Reduced deflection angle:
   DLS
  2  
 E
 c  DOS
2
• Lens equation:
   E
• Source directly behind lens
produces Einstein ring with
angular “radius” E
SIS Lensing
Courtesy of S. Rappaport
SIS with Shear
• Lens equation:
u  x
v y
E x
x y
2
2
E y
x y
2
2
 x
 y
• 1, 2 or 4 images can be produced
SIS with Shear
Courtesy of S. Rappaport
“Double”
Lensing by Galaxies:
Hubble Space Telescope
Images
“Quad”
“Ring”
CASTLES ~ www.cfa.harvard.edu/castles
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
cluster of galaxies,
~1015 Msun
vs.
Clusters look like
this  good!
single galaxy,
~1012 Msun
(Moore et al. 1999)
cluster of galaxies,
~1015 Msun
vs.
single galaxy,
~1012 Msun
(Moore et al. 1999)
Galaxies don’t - bad?
Missing Satellites Problem
Strigari et al (2007)
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):
Rfold
 A   B FA  FB


 Afold d1
 A   B FA  FB
• Flux relation for a cusp triplet (Keeton et al. 2003):
 A   B  C FA  FB  FC
2
Rcusp 

 Acuspd1
 A   B  C FA  FB  FC
• Valid for all smooth mass models
• Deviations  small-scale structure
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 0.006  f  0.07
sub
(Dalal & Kochanek 2002)
• Broadly consistent with CDM
• Is substructure the only viable explanation?
“Minimum Wiggle” Model
• Allow many
multipoles, up to mode
kmax
• Models
underconstrained 
large solution space
B2045+265
• Minimize departures
from elliptical
symmetry.
Multipole Formalism
• Convergence:
1
 (r , )  G ( )
2r
• Lens potential:
 (r , )  rF ( )
• 2-D Poisson equation:
   2
2
 G( )  F ( )  F '' ( )
• Fourier expansion:
k max
a0
F ( )   [ak cos(k )  bk sin( k )]
2 k 1
Observational Constraints
   
• Lens equation: U  X   (X)
• Image positions give 2n constraints

 U 
1
• Magnification:   det   
 X 


• Flux ratios give n-1 constraints
• Combine 3n-1
into a single matrix

 constraints

equation:
Aχ  b
Solving for Unknowns
•Use SVD to solve for parameters when kmax>4:

 (0)

 (i )
  ci
i
•Minimize departure from elliptical symmetry
(i.e., minimize wiggles):
r
2

8
 1  k  a
k max
1
2
k 3
2 2
2
k
b
2
k

•Adding shear leads to nonlinear equations
Solution for B2045+265
Solution for B2045+265
kmax  9
Isodensity contours (solid)
and critical curves (dashed)
kmax  17
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 with
Fold and Cusp Configurations
Submitted to J. Math. Phys.
Lens Time Delays
Robust probe of dark
matter substructure?
Q0957+561
Kundić et al. (1997)
Local Coordinates
Caustic (source plane)
fold
cusp
Critical curve (image plane)
Perturbation Theory for Fold Lenses
• Lens Potential:
1
1 2
2
 (1 , 2 )  (1  K )1   2  e13  f12 2  g1  22  h 23
2
2
 k14  m13 2  n12 22  p1  23  r 24
• Lens Equation:

u1  1 
1

u2   2 
 2
• For small displacements: (u1, u2)
 (eu1, eu2)
• Expand image positions in ε:
1  1e 1/ 2  1e  O(e )3 / 2
 2   2e 1/ 2   2e  O(e )3 / 2
• Solve for coefficients to find:
3hu1  gu2
1 
e  O(e )3 / 2
3hK
 u2 1/ 2 3ghu1  g 2u2

3/ 2
2  
e 
e

O
(
e
)
3h
9h 2 K

• Image separation:
 4u2 1/ 2
d1 
e  O(e )3 / 2
3h
Time-Delay Relation for Fold Pairs
• Scaled Time Delay:


1
  1  eu1 2   2  eu2 2  1 , 2 
2
• Use perturbation theory to get differential time delay:
16
h 3
3/ 2
2
 fold        
(eu2 )  O(e )   d1
27h
2
• Time-delay anomalies may provide a more sensitive
probe of small-scale structure than flux-ratio
anomalies
Comparison to “Exact” Numerical Solution
E
E
Analytic scaling is astrophysically relevant
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 h
along Caustic
Variation of h
along Caustic
Variation of h
along Caustic
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 fourimage 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
HE 0230-2130
B1608+656
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