A “Multimessenger” Approach to Substructure Lensing

45  Download (0)

Full text

(1)

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

(2)

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?

(3)

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.

(4)

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

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

(5)

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

Finite source effects: Observations

Heuristically, each wavelength probes substructure above some mass threshold:

δµ(λ) ∼ Z mhi

m(Rsrc(λ))

m dN dm dm Plus, useful “resonance” if Rein(m) ≈ Rsrc. Many possibilities:

I radio

I near-IR

I optical emission lines

I optical continuum

I X-ray

Of course, need mapping λ ↔ Rsrc.

(6)

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

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)

(7)

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

Time delay millilensing

(CRK & Moustakas 2009)

(8)

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

Time delay millilensing

(CRK & Moustakas 2009)

(9)

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

Time delay millilensing

(CRK & Moustakas 2009)

(10)

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

Time delay millilensing

(CRK & Moustakas 2009)

(11)

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

Time delay millilensing

(CRK & Moustakas 2009)

(12)

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

Random time delays

(13)

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

Scalings

σt∝ fs

m2 hmi

!1/2

(14)

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

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.

(15)

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

(Congdon, CRK & Nordgren ApJ submitted)

(16)

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

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.

(17)

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

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)

(18)

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

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?

(19)

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

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) d3Nc Further marginalize over smooth model parameters:

P (s) ∝ Z

P (s, q | d) dq

(20)

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

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, (γc, γs) at each image position.

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

(21)

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

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

φ = X

i

mi

π ln ri

 αx

αy



= −X

i

mi

πri

 cos θi

sin θi



 γc γs



= −X

i

mi πri2

 cos 2θi sin 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 handle those, we can use CLT on the bulk of the remaining population.

(22)

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

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

φ = X

i

mi

π ln ri

 αx

αy



= −X

i

mi

πri

 cos θi

sin θi



 γc γs



= −X

i

mi πri2

 cos 2θi sin 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 handle those, we can use CLT on the bulk of the remaining population.

(23)

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

Clump population

N clumps.

I position: xi

I “mass” [with dimension of area]: mi= Micrit

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

Then probabilities multiply:

ptot(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:

κs(x) = N Z

p(x, m) m dm ⇒ κs(x) = N hmi px(x)

(24)

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

Local analysis: Uniform case

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

Work through simple case analytically for illustration.

Uniform probability:

px(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:

γi= mi πri2

What is the probability distribution for the largest shear?

(25)

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

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

Pi(> γ) = Z

mi πr2i

px(xi) pm(mi) d2xi dmi

= 1

N hmi Z

dm pm(m) Z

Z (πγm)1/2

0

dr r κs

= 1

N hmi Z

dm pm(m) × 2π × 1 2

 m πγ

 κs

= κs 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!

(26)

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

Deflection

Deflection strength:

αi= mi πri

What is the probability distribution for the largest deflection?

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

Pi(> α) = Z

mi πri

px(xi) pm(mi) d2xi dmi

= 1

N hmi Z

dm pm(m) Z

dθ Z παm

0

dr r κs

= κs

N πα2 m2

hmi

Probability that all deflections are smaller than α:

Pall(< α) = 1 − κs N πα2

m2 hmi

!N

→ exp − κs πα2

m2 hmi

!

(27)

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

Example: Uniform case

Basic population parameters:

κs = 0.01 Rmax = 10000

N = 109189 Mass function:

dN

dM ∝ M−1.8 (M1< M < M2) hM i = 108M

M2

M1 = 100

(28)

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

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

(29)

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

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

(30)

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

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

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)

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

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

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

potential (arcsec^2)

(32)

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

Long-range analysis

Illustrate with deflection:

αx=X

i

αxi

Variance:

var(αx) = α2x − hαxi2

= X

i

X

j

xiαxji − X

i

xii

!2

= X

i

α2xi +X

i

X

j6=i

xii hαxji −

N hαxii2

= Nα2xi − N hαxii2

(33)

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

xii = 1 N hmi

Z

dmi pm(mi) Z

d2xi κs(xi) micos θi

πri

α2xi

= 1

N hmi Z

dmi pm(mi) Z

d2xi κs(xi)  micos θ πri

2

Note

N hαxii2∼ O 1 N



Thus

var(αx) ≈ Nα2xi = 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?

(34)

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

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)

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

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)

(36)

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

Local ortotal.

Potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

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

PDF

potential (arcsec^2)

(37)

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

Effects of mass function

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

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

(38)

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

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

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)

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

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

Total potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

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

PDF

potential (arcsec^2)

(40)

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

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)

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

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

PDF

deflection (arcsec)

(42)

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

Uniformvs.isothermal.

Total potential:

0 0.2 0.4 0.6 0.8 1

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

PDF

potential (arcsec^2)

(43)

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

Lensing Complementarity

Scaled distance, ˆr = r/Rein.

observable mnemonic mass scale spatial scale fluxes δµ ∼ 1/ˆr2 none quasi-local positions δx ∼ Rein/ˆr m2 / hmi intermediate time delays δt ∼ R2einln 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.

(44)

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

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)

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

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)

Figure

Updating...

References

Related subjects :