A ``Multimessenger'' Approach to Substructure...

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

A “Multimessenger” Approach toSubstructure Lensing

Chuck Keeton

July 22, 2009

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Substructure lensing

Past.

I Mao & Schneider 1998: connect flux ratios and substructure

I Metcalf & Madau 2001, Chiba 2002: propose to test CDM

I Dalal & Kochanek 2002: constrain substructure mass fraction

Present/future.

I New observables.

I Can we learn more about substructure?

I Is there really a population of clumps?

I Can we constrain its: mass function? spatial distribution?redshift evolution?

I What does it reveal about dark matter?

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Flux ratio anomalies

without clump with clump

Flux ratio anomalies are hard to miss, and hard to misinterpret.(CRK, Gaudi & Petters 2003, 2005; Congdon & CRK 2005; Yoo et al. 2005, 2006)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

B1555+375

Marlow et al. (1999)

Smooth models generically predict A−B ≈ 0. (CRK, Gaudi & Petters 2005)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

B2045+265

Fassnacht et al. (1999)

Smooth models predict A−B + C ≈ 0. (CRK, Gaudi & Petters 2003)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Flux ratios and substructure

Dalal & Kochanek (2002): constrain substructure mass fraction

0.001 0.01 0.1 1

Single-wavelength flux ratios reveal∫m

dN

dmdm

but say little about dN/dm itself.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Finite source effects

Flux perturbation depends on size of subhalo relative to size ofsource. (Dobler & CRK 2006)

-2

-1

0

1

2

-2 -1 0 1 2

a = 1.2

-2

-1

0

1

2

-2 -1 0 1 2

a = 1.2

-2

-1

0

1

2

-2 -1 0 1 2

a = 1.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 1.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 1.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 1.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 1.2

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook



-2

-1

0

1

2

-2 -1 0 1 2

a = 0.8

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.8

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.8

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook



-2

-1

0

1

2

-2 -1 0 1 2

a = 0.4

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.4

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.4

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.4

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.4

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.4

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.4

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook



-2

-1

0

1

2

-2 -1 0 1 2

a = 0.2

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.2

-2

-1

0

1

2

-2 -1 0 1 2

a = 0.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.2

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

a = 0.2

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Finite source effects(Dobler & CRK 2006)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

0.001 0.01 0.1 1 10

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0.001 0.01 0.1 1 10 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

0.001 0.01 0.1 1 10

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Multiwavelength flux ratios

Heuristically, each wavelength probes substructure above somemass threshold:

δµ(λ) ∼∫ mhi

m(Rsrc(λ))

mdN

dmdm

Plus, useful “resonance” if Rein(m) ≈ Rsrc.

Many possibilities:

I radio

I mid-IR (Chiba et al. 2005, Poindexter et al. 2007, Minezaki et al. 2009, Fadely & CRK)

I optical emission lines (Moustakas & Metcalf 2003, Metcalf et al. 2004, CRK et al. 2006, Sluse et

al. 2007, Sugai et al. 2007, Eigenbrod et al. 2008)

I optical continuum

I X-ray (Blackburne et al. 2006, Pooley et al. 2006, 2007, 2009, Kochanek et al. 2007, Morgan et al. 2008,

Chartas et al. 2009, Dai et al. 2009)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass “threshold”

Band corresponds to range of redshifts in current sample.

1 100 104 106 108M HMsunL

1017

1018

1019

1020

1021

Rein HcmL

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Milli-imagesMG 2016+112, Koopmans et al. (2002)

Center at RA 20 19 18.04171 DEC 11 27 11.3350

BOTH: 2016+112 IPOL 4984.990 MHZ 2016R3U804K.ICLN02.2PLot file version 2 created 01-NOV-2000 16:54:49

Grey scale flux range= 0.000 6.000 MilliJY/BEAMCont peak flux = 6.6350E-03 JY/BEAM Levs = 1.410E-04 * (-3, 3, 5, 10, 20)

Mill

iAR

C S

EC

MilliARC SEC100 50 0 -50 -100

50

40

30

20

10

0

-10

-20

-30

Center at RA 20 19 18.18469 DEC 11 27 14.5700

BOTH: 2016+112 IPOL 4984.990 MHZ 2016R3U804K.ICLN.2PLot file version 1 created 01-NOV-2000 17:02:38


Mill

iAR

C S

EC

MilliARC SEC30 20 10 0 -10 -20 -30 -40

30

20

10

0

-10

-20

-30

-40

Center at RA 20 19 17.97967 DEC 11 27 13.0600

BOTH: 2016+112 IPOL 4984.990 MHZ 2016R3U804K.ICLN01.2PLot file version 1 created 01-NOV-2000 17:01:09


Mill

iAR

C S

EC

MilliARC SEC40 30 20 10 0 -10 -20 -30

30

20

10

0

-10

-20

-30

-40

B2B1

C2

C13C12C11

A1A2

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Milli-imagesB0128+437, Biggs et al. (2004)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Milli-imagesMG J0414+0534, Trotter et al. (2000)

Mill

iAR

C S

EC

MilliARC SEC200 150 100 50

500

480

460

440

420

400

380

360

340

320

Mill

iAR

C S

EC

MilliARC SEC-500 -550 -600 -650

2000

1950

1900

1850

Mill

iAR

C S

EC

MilliARC SEC100 50 0 -50 -100

100

50

0

-50

-100

Mill

iAR

C S

EC

MilliARC SEC-1850 -1900 -1950 -2000

400

350

300

250

200

p

q

s

r

s

r

qp

r

sp

q

sr

q

p

A2

A1

B

C

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Gravitational imaging

Vegetti & Koopmans (2009ab): clumps & 108M� visibly perturbEinstein ring images (a la SLACS)

(my illustration)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Time delay millilensing

(CRK & Moustakas 2009)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Random time delays

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Scalings

σt ∝

(fs

⟨m2⟩

〈m〉

)1/2

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

RX J1131−1231Morgan et al. (2006): M2 (2.2± 1.6 d) M1 (9.6± 2.0 d) S1.

Smooth models predict M1 leads M2.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

RX J1131−1231Morgan et al. (2006): M2 (2.2± 1.6 d) M1 (9.6± 2.0 d) S1.

But substructure can reverse that ordering. (CRK & Moustakas 2009)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Summary of observables

I radio flux ratios

I multiwavelength flux ratios

I milli-images

I gravitational imaging

I time delays

I . . . ?

They probe different aspects of the subhalo population . . .but how, exactly?

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Lensing with stochastic substructure

Can we develop a general theory of substructure lensing?

1. Improve substructure modeling:I fasterI richer — broader substructure modelsI better — understanding of systematic uncertainties

2. Develop general insights:I how is information about substructure encoded in lensing

observables?I what are the “reduced observables” we can/should aim to

measure?

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Framework

Lensing potential from 2-d Poisson equation:

∇2φ = 2κ = 2Σ

Σcrit

Time delay:

τ(x;u) =1 + zlc

DlDs

Dls

[12|x− u|2 − φ(x)

]Fermat’s principle ∇xτ = 0 gives lens equation:

u = x−α(x) where α(x) = ∇φ(x)

Distortions/magnifications:

M =(∂u

∂x

)−1

=[

1− φxx −φxy−φxy 1− φyy

]−1

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Characterizing substructure effects

Can express observables in terms of

potential φ

deflection αx =∂φ

∂x

αy =∂φ

∂y

convergence κ =12

(∂2φ

∂x2+∂2φ

∂y2

)shear γc =

12

(∂2φ

∂x2− ∂2φ

∂y2

)γs =

∂2φ

∂x∂y

Formally, we need to know the (joint) probability distribution ofΦ = {φ, αx, αy, κ, γc, γs} at all image positions.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Deflection

αx =∂φ

∂xαy =

∂φ

∂y

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Convergence

κ =12

(∂2φ

∂x2+∂2φ

∂y2

)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Shear

γc =12

(∂2φ

∂x2− ∂2φ

∂y2

)γs =

∂2φ

∂x∂y

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Use polar coordinates (ri, θi) centered on an image:

φ =∑i

mi

πln ri[

αxαy

]= −

∑i

mi

πri

[cos θisin θi

][γcγs

]= −

∑i

mi

πr2i

[cos 2θisin 2θi

](Small corrections for any clump that overlaps line of sight.)

Can we just use the Central Limit Theorem?

No: variances diverge.

Trouble caused by clump(s) closest to image. If we can handlethose, we can use CLT on the bulk of the remaining population.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Use polar coordinates (ri, θi) centered on an image:

φ =∑i

mi

πln ri[

αxαy

]= −

∑i

mi

πri

[cos θisin θi

][γcγs

]= −

∑i

mi

πr2i

[cos 2θisin 2θi

](Small corrections for any clump that overlaps line of sight.)

Can we just use the Central Limit Theorem?No: variances diverge.

Trouble caused by clump(s) closest to image. If we can handlethose, we can use CLT on the bulk of the remaining population.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Local analysis: Uniform spatial distribution

Goal: find probability distributions for most extreme shear,deflection, and potential.

Work through uniform case analytically for illustration.

px(x) =κs

N 〈m〉

(Over some large but finite area such that∫px(x) d2x = 1.)

Polar coordinates (ri, θi) centered on an image. Shear strength:

γi =mi

πr2i

What is the probability distribution for the largest shear?

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Probability that the shear from clump i is bigger than γ:

Pi(>γ) =∫miπr2i

>γ

px(xi) pm(mi) d2xi dmi

=1

N 〈m〉

∫dm pm(m)

∫dθ

∫ ( mπγ )1/2

0

dr r κs

=1

N 〈m〉

∫dm pm(m)× 2π × 1

2

(m

πγ

)κs

=κsNγ

Probability that all shears are smaller than γ:

Pall(<γ) =(

1− κsNγ

)N→ exp

(−κsγ

)for N →∞

This is the cumulative probability distribution for γmax.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Deflection strength:

αi =mi

πri

What is the probability distribution for the largest deflection?

Probability that the deflection from clump i is bigger than α:

Pi(>α) =∫miπri

>α

px(xi) pm(mi) d2xi dmi

=1

N 〈m〉

∫dm pm(m)

∫dθ

∫ mπα

0

dr r κs

=κs

Nπα2

⟨m2⟩

〈m〉

Probability that all deflections are smaller than α:

Pall(<α) =

(1− κs

Nπα2

⟨m2⟩

〈m〉

)N→ exp

(− κsπα2

⟨m2⟩

〈m〉

)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Local analysis: Power law spatial dist’n

Substructure population: κs ∝ rη−2 with 0 < η ≤ 2.

Taylor series expansions for large local shear/deflection:

P (<γ) = 1− κs,img

γ

+κs,img

γ2

[κs,img

2−⟨m2⟩

〈m〉(η − 2)2

8πr2img

]+O

(γ−3

)P (<α) = 1−

⟨m2⟩

〈m〉κs,img

πα2

+κs,img

2π2α4

[⟨m2⟩2

〈m〉2κs,img −

⟨m4⟩

〈m〉(η − 2)2

4πr2img

]+O

(α−6

)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Local shear

κs ∝ r−1 and dN/dm ∝ m−1.9.

Fix meff =⟨m2⟩/ 〈m〉. Vary q = mhi/mlo = 1, 10, 100, 1000.

Theory.

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Local shear


Fix 〈m〉. Vary q = mhi/mlo = 1, 10, 100, 1000.

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Local deflection



Theory.

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Local deflection


Fix 〈m〉. Vary q = mhi/mlo = 1, 10, 100, 1000.

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Local potential



0

0.2

0.4

0.6

0.8

1

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002

PD

F

potential (arcsec^2)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Local shear

κs ∝ rη−2 and dN/dm ∝ m−1.9 with q = 100.Isothermal (η = 1), steeper (η = 0.5), and shallower (η = 1.5).

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Local deflection


0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Local potential


0

0.2

0.4

0.6

0.8

1

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002

PD

F


Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Long-range analysis

Can use Central Limit Theorem ⇒ need to know variance.

Illustrate with deflection:

αx =∑i

αxi

Variance:

var(αx) =⟨α2x

⟩− 〈αx〉2

=∑i

∑j

〈αxiαxj〉 −

(∑i

〈αxi〉

)2

=∑i

⟨α2xi

⟩+∑i

∑j 6=i

〈αxi〉〈αxj〉 −(N 〈αxi〉

)2

= N⟨α2xi

⟩−N 〈αxi〉2

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

〈αxi〉 =1

N 〈m〉

∫dmi pm(mi)

∫d2xi κs(xi)

mi cos θiπri⟨

α2xi

⟩=

1N 〈m〉

∫dmi pm(mi)

∫d2xi κs(xi)

(mi cos θπri

)2

Note

N 〈αxi〉2 ∼ O(

1N

)Thus

var(αx) ≈ N⟨α2xi

⟩=

⟨m2⟩

〈m〉

∫d2xi κs(xi)

(cos θπri

)2

Similar analysis for all quantities.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Local vs. total: Shear

κs ∝ r−1 and dN/dm ∝ m−1.9 with q = 100.

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Local vs. total: Deflection


0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Local vs. total: Potential


0

0.2

0.4

0.6

0.8

1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

PD

F


Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Total shear



0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Total deflection



0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Mass function: Total potential



0

0.2

0.4

0.6

0.8

1

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

PD

F


Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Total shear


0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

PD

F

shear

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Total deflection


0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

PD

F

deflection (arcsec)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Spatial distribution: Total potential


0

0.2

0.4

0.6

0.8

1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

PD

F


Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Lensing Complementarity

Einstein radius, Rein ∝ m1/2. Scaled distance, r̂ = r/Rein.

observable mnemonic mass scale spatial scale

magnifications δµ ∼ 1/r̂2∫m dN

dm dm quasi-local

positions δx ∼ Rein/r̂⟨m2⟩/ 〈m〉 intermediate

time delays δt ∼ R2ein ln r̂

⟨m2⟩/ 〈m〉 long-range

Different observables contain different information about theclump population.

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Outlook

what how reduced observables

mass function combine∫m (dN/dm) dm

observables and⟨m2⟩/ 〈m〉

low-mass cutoff multiwavelength∫m (dN/dm) dm for

(e.g., WDM) flux ratios different thresholds

spatial time delays something likedistribution

∫r−n κs(x) d2x

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Outlook

Radio quads.

I radio loud: 9 — 3 currently have milli-images

I radio quiet: &20 — doable with EVLA and e-MERLIN?(N. Jackson, O. Wucknitz)

Multiwavelength (optical/IR).

I Others: 4 quads + 1 double published

I Fadely: 1 quad + 5 doubles now, 2 quads + 8 doubles soon

Quad time delays. (Congdon et al. ApJ submitted)

I 7 known currently; more and better measurements to come. . .

I 1 with clear evidence for substructure

I 4 with “anomalies”

Gravitational imaging: &100 SLACS lenses. (S. Vegetti et al.)

Observables

Flux ratios

Multiwavelength

Milli-images

Grav. imaging

Time delays

Recap

Theory

Framework

Local

Long-range

Local vs. total

Total

Complementarity

Outlook

Outlook

Wide-field time-domain surveys will yield thousands of new lenses:Pan-STARRS, DES, LSST, SKA, . . .

Dream:

I Clean sample of ∼100 quads.

I Radio/mid-IR photometry: multiwavelength flux ratios.

I Radio interferometry: milli-images.

I Optical/near-IR monitoring: precise time delays.(Also microlensing and AGN structure.)

The Observatory for Multi-Epoch Gravitational lens Astrophysics(OMEGA)! (L. Moustakas et al.)

A ``Multimessenger'' Approach to Substructure...

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