How can Mathematics Reveal? Chuck Keeton Rutgers University April 2, 2010
Evidence for dark matter galaxy dynamics clusters of galaxies (dynamics, X-rays) large-scale structure cosmography gravitational lensing Big Bang Nucleosynthesis...
Dark matter is... everywhere clustered cold and collisionless (0 th order) not stars, planets, gas,... ( baryonic matter) believed to be an exotic particle ( non-baryonic ) WIMP SuperWIMP sterile neutrino axion hidden sector...
Dark matter is clustered Left: Via Lactea 2 (Diemand et al. 2008) Right: Aquarius Project (Springel et al. 2008)
Missing satellites problem (Strigari et al. 2007)
Astrophysics of galaxy formation Whether subhalos light up depends on: photoevaporation 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)
Physics of dark matter Various candidates all compatible with large-scale structure. Possible suppresion of small-scale structure. (Gao & Theuns 2007; also Coĺın et al. 2000; Bode et al. 2001; Davé et al. 2001; Zentner & Bullock 2003)
Studying dark matter substructure... Tests CDM predictions. 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 time evolution of DM substructure in galaxies.
Basic optics
Gravitational optics
Gravitational optics
Gravitational lensing http://chandra.harvard.edu/photo/2003/apm08279/more.html
2-image lensing Spherical lens. source plane image plane Einstein radius: θ E = 4GM c 2 D ls D ol D os
Einstein ring Spherical lens. source plane image plane Einstein radius: θ E = 4GM c 2 D ls D ol D os
4-image lensing Ellipsoidal lens. source plane image plane
Hubble Space Telescope images (CASTLES project, http://www.cfa.harvard.edu/castles)
Lens time delays Lens Time Delays Q0957+561 Kundic et al.!1997, ApJ, 482, 75" (Kundic et al. 1997)
Key theory Effectively just 2-d gravity. Projected and scaled potential: Time delay: τ(x; u) = 1 + z l c 2 φ = 2 Σ Σ crit D l D s D ls [ ] 1 2 x u 2 φ(x) Fermat s principle x τ = 0 gives lens equation: Distortions/magnifications: M = u = x φ(x) ( ) 1 [ u 1 φxx φ = xy x φ xy 1 φ yy ] 1
Fermat s principle Time delay surface: [ ] 1 τ(x; u) = τ 0 2 x u 2 φ(x) 1
Astrophysical applications As a tool, gravitational lensing can be applied to diverse problems. dark matter in and around galaxies galaxy masses and evolution galaxy environments cosmological parameters quasar structure extrasolar planets/asteroids black holes as astrophysical objects black holes as relativistic objects theories of gravity braneworld model
Basic image counting For a typical galaxy, expect 2 or 4 bright images. 4-image lenses come in 3 basic configurations:
Maximum number of images? Explicit construction: 4/6/8 images from a galaxy whose density is constant on similar ellipses, plus tidal forces from neighboring galaxies. (CRK, Mao & Witt 2000)
Exotic lenses PMN J0134 0931: 5 images of a quasar in an unexpected configuration (plus at least 1 image of a second source). There must be two lens galaxies. (Winn et al. 2002, 2003; CRK & Winn 2003)
Being rigorous Would be nice to have rigorous results for: spherical or ellipsoidal mass distributions different density profiles with or without tidal shear 1, 2,... galaxies etc. (cf. A. Eremenko)
Odd image theorem Burke (1981) used the Poincaré-Hopf index theorem to argue: A transparent galaxy, not necessarily spherical, produces an odd number of images. (Assumes the deflection is bounded.)
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)
Supermassive black holes Smooth galaxy: always 1 central image With SMBH (point mass) at the center: 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 more than one SMBH? What can we say about the number of images and their properties? (cf. D. Khavinson)
Flux ratio anomalies Easy to explain image positions (even to 0.1% precision): ellipsoidal galaxy tidal forces from environment But hard to explain flux ratios! expected observed (Marlow et al. 1999)
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)
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)
Anomalies are ubiquitous (Credits: Fassnacht et al. 1999; Marlow et al. 1999; Pooley et al. 2006ab)
Universal relations fold: µ A + µ B 0 cusp: µ A + µ B + µ C 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 substructure (Diemand et al. 2008)
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)
(CRK & Moustakas 2009)
(CRK & Moustakas 2009)
(CRK & Moustakas 2009)
(CRK & Moustakas 2009)
(CRK & Moustakas 2009)
(CRK & Moustakas 2009)
Stochasticity
Astrophysical import Dalal & Kochanek (2002) analyzed flux ratios in 7 quad lenses: mean substructure mass fraction f sub 0.02 (0.006 0.07 at 90% confidence) Digging deeper. New observables. Can we learn more about substructure? Is there really a population of clumps? Can we constrain its: mass function? spatial distribution? time evolution? What does it reveal about dark matter?
Framework Each clump has some random mass m i and position (r i, θ i ). Potential: φ = m i π ln r i i Deflection: [ αx ] = α y i m i [ cos θi ] πr i sin θ i Tidal shear: [ γc γ s ] = i m i πri 2 [ cos 2θi sin 2θ i ] Treat as a stochastic process, compute (joint) probability densities. (cf. A. Teguia)
Lensing complementarity How do different lensing observables depend on the population of dark matter clumps? observable mass scale spatial scale fluxes positions time delays m pm (m) dm m 2 p m (m) dm m 2 p m (m) dm quasi-local intermediate long-range (CRK 2009)
Image counting Lens equation (vector form): [ κ + γ 0 u = x 0 κ γ ] x i m i π x x i x x i 2 Roots of a random polynomial! (work by An, Evans, Khavinson, Neumann, Petters, Rhie,...)
Gravitational lensing is rich in both astrophysics and mathematics. Great synergy. Work together to learn about dark matter!