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
Universal Relations
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
Gravitational Lensing
Image Counting
Universal Relations
Stochastic Lensing
Summary
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
Universal Relations
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, . . .)
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