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