A “Multimessenger” Approach to Substructure Lensing

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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

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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?

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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.

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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

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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.

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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)

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Time delay millilensing

(CRK & Moustakas 2009)

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Time delay millilensing

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Time delay millilensing

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Time delay millilensing

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Time delay millilensing

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Random time delays

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Scalings

σt∝ fs

m² hmi

!1/2

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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.

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(Congdon, CRK & Nordgren ApJ submitted)

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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.

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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)

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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?

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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^3Nc Further marginalize over smooth model parameters:

P (s) ∝ Z

P (s, q | d) dq

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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}).

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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_i²

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.

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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_i²

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.

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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(x₁, m₁, x₂, m₂, . . .) = p(x₁, m₁) × p(x₂, m₂) × . . . 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)

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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) d²x = 1.) Polar coordinates (ri, θi) centered on an image. Shear strength:

γ_i= m_i πr_i²

What is the probability distribution for the largest shear?

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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) d²x_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!

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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) d²x_i dm_i

= 1

N hmi Z

dm pm(m) Z

dθ Z _πα^m

0

dr r κs

= κs

N πα² m²

hmi

Probability that all deflections are smaller than α:

P_all(< α) = 1 − κ_s N πα²

m² hmi

!^N

→ exp − κ_s πα²

m² hmi

!

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Example: Uniform case

Basic population parameters:

κ_s = 0.01 Rmax = 100⁰⁰

N = 109189 Mass function:

dN

dM ∝ M^−1.8 (M1< M < M2) hM i = 10⁸M

M2

M₁ = 100

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Example: Isothermal case

Basic population parameters:

κs(r) = 0.01 r Rmax = 100⁰⁰

N = 1091 Mass function:

dN

dM ∝ M^−1.8 (M₁< M < M₂) hM i = 10⁸M

M₂ M1

= 100

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Effects of mass function

Fiducial mass function,same hmi, or same meff =m² / 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

PDF

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

PDF

shear

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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

PDF

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

PDF

deflection (arcsec)

(31)

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

PDF

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

PDF

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Long-range analysis

Illustrate with deflection:

αx=X

i

αxi

Variance:

var(α_x) = α²_x − hαxi²

= X

i

X

j

hαxiαxji − X

i

hαxii

!²

= X

i

α²_xi +X

i

X

j6=i

hαxii hαxji −

N hαxii2

= Nα²_xi − N hαxii²

(33)

hαxii = 1 N hmi

Z

dmi pm(mi) Z

d²xi κs(xi) micos θi

πri

α²_xi

= 1

N hmi Z

dm_i p_m(m_i) Z

d²x_i κ_s(x_i) m_icos θ πri

²

Note

N hαxii²∼ O 1 N

Thus

var(αx) ≈ Nα²_xi = m² hmi

Z

d²xi κs(xi) cos θ πri

2

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

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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

PDF

shear

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08 0.1

PDF

shear

(35)

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

PDF

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

PDF

deflection (arcsec)

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Local ortotal.

Potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

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

PDF

(37)

Effects of mass function

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

PDF

shear

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08 0.1

PDF

shear

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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

PDF

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

PDF

deflection (arcsec)

(39)

Total potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

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

PDF

(40)

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

PDF

shear

(41)

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

PDF

deflection (arcsec)

(42)

Total potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

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Lensing Complementarity

Scaled distance, ˆr = r/R_ein.

observable mnemonic mass scale spatial scale fluxes δµ ∼ 1/ˆr² none quasi-local positions δx ∼ Rein/ˆr m² / hmi intermediate time delays δt ∼ R²_einln ˆr m² / 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.

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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

(45)

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)

A “Multimessenger” Approach to Substructure Lensing

Full text

A “Multimessenger” Approach to Substructure Lensing

Millilensing: A past, a present, a future?

Flux ratios

Finite source effects: Theory

Finite source effects: Observations

New observables?

Time delay millilensing

Time delay millilensing

Time delay millilensing

Time delay millilensing

Time delay millilensing

Random time delays

Scalings

Time delay anomalies

RX J1131−1231

RX J1131−1231

Lensing with stochastic substructure

Substructure modeling

Let’s think for a moment

Clump population

Local analysis: Uniform case

Deflection

Example: Uniform case

Example: Isothermal case

Effects of mass function

Long-range analysis

Local vs. total

Effects of mass function

Effects of spatial distribution

Lensing Complementarity

Work in progress

Biases?

Figure

References

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