Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

SummaryHow can Mathematics Reveal

Dark Matter?

Chuck Keeton

Rutgers University

April 2, 2010

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Evidence for dark matter

I galaxy dynamics

I clusters of galaxies (dynamics, X-rays)

I large-scale structure

I cosmography

I gravitational lensing

I Big Bang Nucleosynthesis

I . . .

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Dark matter is . . .

I everywhere

I clustered

I “cold” and “collisionless” (0th order)

I not stars, planets, gas, . . . (“baryonic” matter)

I believed to be an exotic particle (“non-baryonic”)I WIMPI SuperWIMPI sterile neutrinoI axionI hidden sectorI . . .

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Dark matter is clustered

Left: Via Lactea 2 (Diemand et al. 2008)Right: Aquarius Project (Springel et al. 2008)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

“Missing satellites” problem

(Strigari et al. 2007)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Astrophysics of galaxy formationWhether subhalos “light up” depends on:

I photoevaporation

I efficiency of star formation

(Strigari et al. 2007; also Bullock et al. 2000; Taylor & Babul 2001, 2004; Somerville 2002; Benson et al. 2002; Zentner et al.

2003, 2005; Koushiappas et al. 2004; Kravtsov et al. 2004; Oguri & Lee 2004; van den Bosch et al. 2005)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Physics of dark matter

Various candidates — all compatible with large-scale structure.

Possible suppresion of small-scale structure.

8

(Gao & Theuns 2007; also Colın et al. 2000; Bode et al. 2001; Dave et al. 2001; Zentner & Bullock 2003)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Studying dark matter substructure . . .

Tests CDM predictions.

I Do “dark dwarfs” exist?

Probes the astrophysics of galaxy formation on small scales.

Provides astrophysical evidence about the nature of dark matter.

Goal: Measure mass function, spatial distribution, and timeevolution of DM substructure in galaxies.

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Basic optics

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

“Gravitational” optics

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Gravitational “optics”

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Gravitational lensing

http://chandra.harvard.edu/photo/2003/apm08279/more.html

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

2-image lensing

Spherical lens.

source plane image plane

Einstein radius: θE =√

4GMc2

Dls

DolDos

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Einstein ring

Spherical lens.

source plane image plane

Einstein radius: θE =√

4GMc2

Dls

DolDos

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

4-image lensing

Ellipsoidal lens.

source plane image plane

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Hubble Space Telescope images

(CASTLES project, http://www.cfa.harvard.edu/castles)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Lens time delays

Lens Time Delays

• Q0957+561Kundic et al. !1997, ApJ, 482, 75"

(Kundic et al. 1997)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Key theory

Effectively just 2-d gravity. Projected and scaled potential:

∇2φ = 2Σ

Σcrit

Time delay:

τ(x; u) =1 + zl

c

DlDs

Dls

[12|x− u|2 − φ(x)

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

u = x−∇φ(x)

Distortions/magnifications:

M =(∂u

∂x

)−1

=[

1− φxx −φxy

−φxy 1− φyy

]−1

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Fermat’s principle

Time delay surface: τ(x; u) = τ0

[12|x− u|2 − φ(x)

]

– 1 –

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Astrophysical applications

As a tool, gravitational lensing can be applied to diverse problems.

I dark matter in and around galaxies

I galaxy masses and evolution

I galaxy environments

I cosmological parameters

I quasar structure

I extrasolar planets/asteroids

I black holes as astrophysical objects

I black holes as relativistic objects

I theories of gravity — braneworld model

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Basic image counting

For a typical galaxy, expect 2 or 4 bright images.

4-image lenses come in 3 basic configurations:

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Maximum number of images?

Explicit construction: 4/6/8 images from a galaxy whose density isconstant on similar ellipses, plus tidal forces from neighboringgalaxies. (CRK, Mao & Witt 2000)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Exotic lenses

PMN J0134−0931: 5 images of a quasar in an unexpectedconfiguration (plus at least 1 image of a second source).

⇒ There must be two lens galaxies. (Winn et al. 2002, 2003; CRK & Winn 2003)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Being rigorous

Would be nice to have rigorous results for:

I spherical or ellipsoidal mass distributions

I different density profiles

I with or without tidal shear

I 1, 2, . . . galaxies

I etc.

(cf. A. Eremenko)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Odd image theorem

Burke (1981) used the Poincare-Hopf index theorem to argue:

“A transparent galaxy, not necessarily spherical, produces an oddnumber of images.”

(Assumes the deflection is bounded.)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Central images are faint

They are hard to find.

A

C

B

(Winn et al. 2004)

They tell us about the centers of lens galaxies. (e.g., CRK 2003)

Dark Matter

Gravitational Lensing

Image Counting

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Stochastic Lensing

Summary

Supermassive black holes

Smooth galaxy:

I always 1 central image

With SMBH (point mass) at the center:

I either 2 or 0 central images

(Mao et al. 2001)

If we can detect 2 central images, we can measure SMBH masses.(Rusin, CRK & Winn 2005)

But what if the SMBH is not at the center? What if there is morethan one SMBH? What can we say about the number of imagesand their properties? (cf. D. Khavinson)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Flux ratio “anomalies”

“Easy” to explain image positions (even to ∼0.1% precision):

I ellipsoidal galaxy

I tidal forces from environment

But hard to explain flux ratios!

expected observed (Marlow et al. 1999)

Dark Matter

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Stochastic Lensing

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Anomalies are generic

Close pair of images: Taylor series expansion yields

A−B ≈ 0

Universal prediction for smooth models. (CRK, Gaudi & Petters 2005)

(models, CRK et al. 2005; B1555+375, Marlow et al. 1999)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Anomalies are generic

Close triplet of images: Taylor series expansion yields

A−B + C ≈ 0

Universal prediction for smooth models. (CRK, Gaudi & Petters 2003)

(models, CRK et al. 2003; B2045+265, Fassnacht et al. 1999)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Anomalies are ubiquitous

(Credits: Fassnacht et al. 1999; Marlow et al. 1999; Pooley et al. 2006ab)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Universal relations

fold: µA + µB ≈ 0cusp: µA + µB + µC ≈ 0

Can extend to higher-order singularities.(cf. A. Aazami, A. Petters, M. Werner)

Can also apply to lens time delays. (Congdon, CRK & Nordgren 2008, 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Dark matter substructure

(Diemand et al. 2008)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Substructure and lensing

What if lens galaxies contain dark matter clumps?

The clumps can distort the images.

without clump with clump

(cf. Mao & Schneider 1998; Metcalf & Madau 2001; Chiba 2002)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

(CRK & Moustakas 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Stochasticity

Dark Matter

Gravitational Lensing

Image Counting

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Stochastic Lensing

Summary

Astrophysical import

Dalal & Kochanek (2002) analyzed flux ratios in 7 quad lenses:

I mean substructure mass fraction

I fsub ≈ 0.02 (0.006–0.07 at 90% confidence)

Digging deeper.

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?time evolution?

I What does it reveal about dark matter?

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Framework

Each clump has some random mass mi and position (ri, θi).Potential:

φ =∑

i

mi

πln ri

Deflection: [αx

αy

]= −

∑i

mi

πri

[cos θi

sin θi

]Tidal shear: [

γc

γs

]= −

∑i

mi

πr2i

[cos 2θi

sin 2θi

]

Treat as a stochastic process, compute (joint) probability densities.

(cf. A. Teguia)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Lensing complementarity

How do different lensing observables depend on the population ofdark matter clumps?

observable mass scale spatial scale

fluxes∫m pm(m) dm quasi-local

positions∫m2 pm(m) dm intermediate

time delays∫m2 pm(m) dm long-range

(CRK 2009)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Image counting

Lens equation (vector form):

u = x−[κ+ γ 0

0 κ− γ

]x−

∑i

mi

π

x− xi

|x− xi|2

Roots of a random polynomial!

(work by An, Evans, Khavinson, Neumann, Petters, Rhie, . . .)

Dark Matter

Gravitational Lensing

Image Counting

Universal Relations

Stochastic Lensing

Summary

Summary

Gravitational lensing is rich in both astrophysics and mathematics.

Great synergy.

Work together to learn about dark matter!