Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Clump pop’n Local analysis Local examples Long-range Local vs. total Mass function Spatial dist’n Complementarity Biases

### A “Multimessenger” Approach to Substructure Lensing

Chuck Keeton

June 23, 2009

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Clump pop’n Local analysis Local examples Long-range Local vs. total Mass function Spatial dist’n Complementarity Biases

### Millilensing: A past, a present, a future?

Past.

I Mao & Schneider 1998: connect flux ratios and substructure

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

I Dalal & Kochanek 2002: use flux ratios to constrain substructure mass fraction

Present.

I Does lensing need too much substructure? (cf. Chen)

I Can we see it? (cf. Vegetti, Fassnacht)

Future.

I Can we learn more about substructure?

I Is there really a populationsofclumps?

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

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Clump pop’n Local analysis Local examples Long-range Local vs. total Mass function Spatial dist’n Complementarity Biases

### Flux ratios

Flux ratio anomalies are hard to miss, and hard to misinterpret.

(CRK, Gaudi & Petters 2003, 2005; Congdon & CRK 2005; Yoo et al. 2005, 2006)

They reveal the total amount of substructure, Z

m dN dm dm

But single-wavelength flux ratios tell you (essentially) nothing about the mass function.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Finite source effects: Theory

Magnification vs. source size, given an SIS clump. (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

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Finite source effects: Observations

Heuristically, each wavelength probes substructure above some mass threshold:

δµ(λ) ∼
Z m_{hi}

m(R_{src}(λ))

m dN
dm dm
Plus, useful “resonance” if R_{ein}(m) ≈ R_{src}.
Many possibilities:

I radio

I near-IR

I optical emission lines

I optical continuum

I X-ray

Of course, need mapping λ ↔ Rsrc.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### New observables?

Multi-wavelength flux ratios.

I need to know Rsrc(λ)

I need to be careful about variability, microlensing, extinction Image positions.

I can be perturbed at the & 1–10 mas level^{(cf. Chen)}

I but are position shifts unique to substructure? probably not Time delays.

I there is interest in measuring more precise time delays

I might they be sensitive to substructure?

(Also, gravitational imaging . . . cf. Vegetti)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay millilensing

(CRK & Moustakas 2009)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay millilensing

(CRK & Moustakas 2009)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay millilensing

(CRK & Moustakas 2009)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay millilensing

(CRK & Moustakas 2009)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay millilensing

(CRK & Moustakas 2009)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Random time delays

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Scalings

σt∝ fs

m^{2}
hmi

!1/2

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Time delay anomalies

(Congdon, CRK & Nordgren ApJ submitted)

Use realistic smooth models to create mock 4-image lenses.

I Ellipticities and m = 4 multipoles from galaxy photometry.

(Bender et al. 1989, Saglia et al. 1993, Jørgensen et al. 1995)

I Shears from N -body simulations. (Holder & Schechter 2003)

Find mock lenses that match each observed image pair:

I d1= distance between the images

I d2= distance to next-nearest image

I parities

Examine distribution of time delays among the matching mock lenses. Identify outliers as anomalies.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

(Congdon, CRK & Nordgren ApJ submitted)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### RX J1131−1231

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

Smooth models predict M1 leads M2.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### RX J1131−1231

Morgan 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)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Lensing with stochastic substructure

It looks like we want to use multiple observables to probe substructure.

Can we develop a general theory of substructure lensing?

1. Improve substructure modeling:

I faster

I richer — broader substructure models

I better — understanding of systematic uncertainties 2. Develop general insights, e.g.:

I how is information about the substructure population encoded in lensing observables?

I to what extent are flux ratios, image positions, and time delays complementary?

I are there unique signatures of clumpy substructure?

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Substructure modeling

Bayesian framework.

I d, data

I q, smooth model parameters

I s, substructure population parameters

I c = {xi, mi}, clump positions and masses[nuisance!]

Posterior probability, marginalizing over nuisance parameters:

P (s, q | d) ∝ Z

L(d | c, s, q) P (c | s, q) P (s|q) P (q) d^{3N}c
Further marginalize over smooth model parameters:

P (s) ∝ Z

P (s, q | d) dq

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Let’s think for a moment

Can we avoid marginalizing over N clump positions and masses?

(Please?!)

What really matters is Φ = {φ, αx, αy, κ, γc, γs}

I potential, φ

I deflection, (αx, αy)

I convergence, κ

I shear, (γ_{c}, γ_{s})
at each image position.

Dream: analytic probability distribution P ({Φ}).

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Use polar coordinates (r_{i}, θ_{i}) centered on an image:

φ = X

i

mi

π ln ri

αx

αy

= −X

i

mi

πri

cos θi

sin θi

γ_{c}
γ_{s}

= −X

i

m_{i}
πr_{i}^{2}

cos 2θ_{i}
sin 2θ_{i}

(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 handle those, we can use CLT on the bulk of the remaining population.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Use polar coordinates (r_{i}, θ_{i}) centered on an image:

φ = X

i

mi

π ln ri

αx

αy

= −X

i

mi

πri

cos θi

sin θi

γ_{c}
γ_{s}

= −X

i

m_{i}
πr_{i}^{2}

cos 2θ_{i}
sin 2θ_{i}

(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 handle those, we can use CLT on the bulk of the remaining population.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Clump population

N clumps.

I position: xi

I “mass” [with dimension of area]: mi= Mi/Σcrit

Assume clumps are independent and identically distributed (i.i.d.).

Then probabilities multiply:

p_{tot}(x_{1}, m_{1}, x_{2}, m_{2}, . . .) = p(x_{1}, m_{1}) × p(x_{2}, m_{2}) × . . .
If positions and masses are separable:

p(xi, mi) = px(xi) pm(mi)

More intuitive is the mean surface mass density in substructure — what you get if you average over many realizations:

κ_{s}(x) = N
Z

p(x, m) m dm ⇒ κ_{s}(x) = N hmi p_{x}(x)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Local analysis: Uniform case

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

Work through simple case analytically for illustration.

Uniform probability:

p_{x}(x) = κs

N hmi

(Over some large but finite area such thatR px(x) d^{2}x = 1.)
Polar coordinates (ri, θi) centered on an image. Shear strength:

γ_{i}= m_{i}
πr_{i}^{2}

What is the probability distribution for the largest shear?

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Probability that the shear from a given clump is bigger than γ:

P_{i}(> γ) =
Z

mi πr2i

>γ

p_{x}(x_{i}) p_{m}(m_{i}) d^{2}x_{i} dm_{i}

= 1

N hmi Z

dm pm(m) Z

dθ

Z (_{πγ}^{m})^{1/2}

0

dr r κs

= 1

N hmi Z

dm pm(m) × 2π × 1 2

m πγ

κs

= κ_{s}
N γ

Probability that all shears are smaller than γ:

Pall(< γ) =

1 − κs

N γ

N

→ exp

−κs

γ

for N → ∞ This is just the cumulative probability distribution for γmax!

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Deflection

Deflection strength:

αi= m_{i}
πri

What is the probability distribution for the largest deflection?

Probability that the deflection from a given clump is bigger than α:

P_{i}(> α) =
Z

mi πri>α

p_{x}(x_{i}) p_{m}(m_{i}) d^{2}x_{i} dm_{i}

= 1

N hmi Z

dm pm(m) Z

dθ
Z _{πα}^{m}

0

dr r κs

= κs

N πα^{2}
m^{2}

hmi

Probability that all deflections are smaller than α:

P_{all}(< α) = 1 − κ_{s}
N πα^{2}

m^{2}
hmi

!^{N}

→ exp − κ_{s}
πα^{2}

m^{2}
hmi

!

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Example: Uniform case

Basic population parameters:

κ_{s} = 0.01
Rmax = 100^{00}

N = 109189 Mass function:

dN

dM ∝ M^{−1.8} (M1< M < M2)
hM i = 10^{8}M_{}

M2

M_{1} = 100

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Example: Isothermal case

Basic population parameters:

κs(r) = 0.01
r
Rmax = 100^{00}

N = 1091 Mass function:

dN

dM ∝ M^{−1.8} (M_{1}< M < M_{2})
hM i = 10^{8}M_{}

M_{2}
M1

= 100

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Effects of mass function

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

Local shear:

0 0.2 0.4 0.6 0.8 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

shear

0 0.2 0.4 0.6 0.8 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

shear

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

Local deflection:

0 0.2 0.4 0.6 0.8 1

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

deflection (arcsec)

0 0.2 0.4 0.6 0.8 1

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

deflection (arcsec)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

Local potential:

0 0.2 0.4 0.6 0.8 1

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

potential (arcsec^2)

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

potential (arcsec^2)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Long-range analysis

Illustrate with deflection:

αx=X

i

αxi

Variance:

var(α_{x}) = α^{2}_{x} − hαxi^{2}

= X

i

X

j

hαxiαxji − X

i

hαxii

!^{2}

= X

i

α^{2}_{xi} +X

i

X

j6=i

hαxii hαxji −

N hαxii2

= Nα^{2}_{xi} − N hαxii^{2}

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

hαxii = 1 N hmi

Z

dmi pm(mi) Z

d^{2}xi κs(xi) micos θi

πri

α^{2}_{xi}

= 1

N hmi Z

dm_{i} p_{m}(m_{i})
Z

d^{2}x_{i} κ_{s}(x_{i}) m_{i}cos θ
πri

^{2}

Note

N hαxii^{2}∼ O 1
N

Thus

var(αx) ≈ Nα^{2}_{xi} = m^{2}
hmi

Z

d^{2}xi κs(xi) cos θ
πri

2

Same applies to all observables. Question is: are long-range effects significant compared with local effects?

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Local vs. total

Local ortotal.

Shear:

0 0.2 0.4 0.6 0.8 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

shear

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08 0.1

shear

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Local ortotal.

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

deflection (arcsec)

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

deflection (arcsec)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Local ortotal.

Potential:

0 0.2 0.4 0.6 0.8 1

-4 -3 -2 -1 0 1 2 3

potential (arcsec^2)

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

potential (arcsec^2)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Effects of mass function

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

Total shear:

0 0.2 0.4 0.6 0.8 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

shear

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08 0.1

shear

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

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

deflection (arcsec)

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

deflection (arcsec)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Fiducial mass function,same hmi, or same meff =m^{2} / hmi.

Total potential:

0 0.2 0.4 0.6 0.8 1

-4 -3 -2 -1 0 1 2 3

potential (arcsec^2)

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

potential (arcsec^2)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Effects of spatial distribution

Uniformvs.isothermal.

Total shear:

0 0.2 0.4 0.6 0.8 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

shear

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Uniformvs.isothermal.

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

deflection (arcsec)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

Uniformvs.isothermal.

Total potential:

0 0.2 0.4 0.6 0.8 1

-4 -3 -2 -1 0 1 2 3

potential (arcsec^2)

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Lensing Complementarity

Scaled distance, ˆr = r/R_{ein}.

observable mnemonic mass scale spatial scale
fluxes δµ ∼ 1/ˆr^{2} none quasi-local
positions δx ∼ Rein/ˆr m^{2} / hmi intermediate
time delays δt ∼ R^{2}_{ein}ln ˆr m^{2} / hmi long-range

Different observables contain different information about the clump population.

⇒ Even we are frank about systematic uncertainties, we can still expect to learn a lot of new things about substructure.

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Work in progress

Better characterize spatial dependence.

I How many clumps do you need to account for?

Allow “puffy” clumps.

I How many clumps overlap the line of sight?

Line of sight effects? (cf. Chen)

Use this formalism to simplify and enrich substructure modeling.

Stay tuned. . .

I new multi-wavelength flux ratio data

I gravlens 2.0

Flux ratios Finite source Time delays

Millilensing Anomalies General Theory

### Biases?

Are there effects that might bias the amount of substructure found in lens galaxies?

I Shape and inclination. (Mandelbaum, van den Ven & CRK 2009)

I Evironment? (Oguri)

I Line of sight? (Chen)