1. H.-W. Rix, Vatican 2003 Gravitational Lensing as a Tool in Cosmology A Brief History of Lensing 1704 Newton (in Optics): „Do not bodies act upon light at a distance, and bend its rays?“ 1801 Soldner: Are the apparent positions of stars affected by their mutual light deflection? hyperbolic passage with v = c: tan (  /2) = GM/(c 2 r) = R s /(2r) 1911 Einstein: finds the correct General Relativity answer  = 4GM/(c 2 r) = R s /r => differs by factor 2 from the Newtonian value

2. H.-W. Rix, Vatican 2003 1919 Eddington: measures  = 1.6“ at the edge of the sun, confirming General Relativity 1937 Zwicky: galaxies could act as lenses for distant objects - test relativity - magnify distant objects - measure masses 1979 Walsh and Weyman: double quasar 0957 + 561 – first lens!

3. H.-W. Rix, Vatican 2003 Lensing Basics We consider the paths of light in the presence of masses (which curve space). Assumptions: –Minkowski, or FRW, “smooth” space, with localized distortions –Local perturbations are weak –I.e.   C 2 and v source,v lens,v obs << c Fermat’s Principle in gravitational lensing Images are formed at stationary (min,max,saddle) points of the light travel time There are two components to the light travel time: Geometric (detour) delays Relativistic time dilation

4. H.-W. Rix, Vatican 2003 Light Travel Time and Image Formation 1 image 3 images detour Time dilation Total light travel time =source

5. H.-W. Rix, Vatican 2003 Fermat’s principle in Gravitational Lensing (contd.) Relativistic time dilation leads to an effective index of refraction n eff =1+2|  |/c 2 Images are then formed, where is satisfied. Images are formed in pairs  should always expect to see odd number of images View “onto” the sky From Blandford and Narayan 1986

6. H.-W. Rix, Vatican 2003 Lens Equation Simple geometry yields Or Note that this is an implicit equation for  … but, how do we get  ?  = (true) source position  =(seeming) image position  =(scaled) deflection

7. H.-W. Rix, Vatican 2003 Quantifying light deflection Define the “projected gravitational potential”  through  “thin lens” Then the deflection is given by where  is a scaled surface mass density in the lens plane S=source D=deflector O=observer I=image Image deflection is related to the surface mass density in the lens plane

8. H.-W. Rix, Vatican 2003 (Spherically) Symmetric Lenses In the case of a symmetric lens the calculation of  is simple, namely  ~M(<  )/  and the lens equation becomes For a perfect alignment of source and lens, i.e.  =0, image(s) appear at the “Einstein angle”,  e For cosmologically distant objects, lensed by an intervening galaxy, the typical image separations are

9. H.-W. Rix, Vatican 2003 Simple (and important) symmetric lenses Point mass lens Magnification of the images –Gravitational lensing preserves surface brightness  Image amplification comes from area magnification,  for a point source Lens Equation Image positions

10. H.-W. Rix, Vatican 2003 Isothermal Sphere as a Lens The total (stars + dark matter) mass profile is approximately isothermal, i.e.  ~r -2  a simple, but applicable model for galaxies as lenses because M(  )~  the deflection is constant Lens equation:   =  +-   with magnification        image separation  is always 2  E   image separation  is direct measure of the enclosed mass

11. H.-W. Rix, Vatican 2003 Galaxies as (Strong) Lenses Historically the first lenses: “multiple Quasars” Einstein Ring Brown et al 2001 (Walsh and Weyman 1979) PG 1115 Impey et al 1998

12. H.-W. Rix, Vatican 2003 Lensed arcs are magnified pieces of the  QSO host galaxy! Lens Modelling What can we learn from such lens systems –Mass distribution of lens –Structure of source Nature’s telescope –Cosmological parameters, such as H 0 Procedure –Assume lens mass model –Map image back to source –Check match in source plane –Modify lens model  iterate QSO 0957+561 Keeton et al 2002  lens galaxy

13. H.-W. Rix, Vatican 2003 Lens Modeling: Time Delay Light along different image paths takes differently long to reach us. The lens model only determines the fractional difference, typically 10 -10 If we measure time delay in absolute time units  total light travel to source redshift in seconds  H 0 Note: Distance measurement not expansion velocity measurement  independent!

14. H.-W. Rix, Vatican 2003 Time Delay in QSO0957 Kundic et al 1996 (Intrinsic) variability of the image A repeats in the light curve of image B  we are seeing the same object 417 days apart  H 0 =67+-10 km/s/Mpc Note: time delay somewhat model dependent

15. H.-W. Rix, Vatican 2003 Galaxy Mass Estimates from Lensing Observed light from lensing galaxy  luminosity Image separation  galaxy mass  method to measure M/L of galaxies at earlier epochs! From Kochanek, Rix et al. 2000 Evolution of the luminosity at a given mass, compared to models for given (star) formation redshifts  Star formation largely complete by z>2 in massive galaxies

16. H.-W. Rix, Vatican 2003 “Giant Arcs” and Cluster Masses (extended) background galaxy images get highly magnified (tangentially)  arcs  enclosed mass

17. H.-W. Rix, Vatican 2003 Cluster Mass Measurements E.g.: Cluster MS10 from Ettori et al 2001 X-ray and lensing masses agree quite well!

18. H.-W. Rix, Vatican 2003 Nature’s Telescope One distant galaxy in the cluster CL0024 is seen 7 times! Colley et al 2000

19. H.-W. Rix, Vatican 2003 Weak Lensing To get multiple images, one needs a “critical” mass density along the line of sight. However, any mass distribution along the way will distort the image  weak lensing One can describe the lensing in this regime as a linear distortion of the images, i.e. a 2x2 matrix with three independent elements: convergence  and shear  (vector)

20. H.-W. Rix, Vatican 2003 Observable Consequences: Convergence: –Magnification: but we would need to know the source size a priori  difficult Shear: –If all sources were circles: Unique, but very small (few %) ellipticity –But: Sources have much larger intrinsic ellipticity Yet, the position angles of (unrelated) objects should be at random angles  Search for correlated image ellipticities!

21. H.-W. Rix, Vatican 2003 Lensing by Cosmic Large Scale Structure The cosmic large scale structure will create both convergence and shear. We cannot use the “thin-lens” approximation, but must integrate along the line of sight.  Mass structure on small to large scales will cause coherent image distortions. Amplitude and radial dependence of the distortion coherence will depend on “cosmology”  Independent test of large-scale structure

22. H.-W. Rix, Vatican 2003 Lensing Convergence and Shear from Large Scale Structure From White and Hu, 2000 Convergence Field Shear Field

23. H.-W. Rix, Vatican 2003 Measurement and Application of Cosmic Lensing Shear Different measurements of the shear correlation function Resulting constraints on the density  and the fluctuation amplitude  (from Mellier 2003  Note: mass structure estimates without assuming galaxies trace mass

24. H.-W. Rix, Vatican 2003 Galaxy-Galaxy Lensing Projected Mass Overdensity Projected Radius As clusters, individual galaxies distort background images, too. Yet, these distortions are much smaller  Co-add signal from many equivalent (?) galaxies  Galaxy-galaxy lensing signals show that galaxy halos extend far (>200 kpc)

25. H.-W. Rix, Vatican 2003 Is there Halo Sub-Structure? (e.g. Dalal and Kochanek 2001,2002) 1 image 3 images B1555 radio Images A and B should be equally bright! Differential dust extinction? No Micro-lensing by stars? No Halo Sub-structure? ~0.01” image splitting  (de-)magnification

26. H.-W. Rix, Vatican 2003 Dalal and Kochanek 2002 How much do the observed image brightnesses deviate from the best smooth model fit? Halo sub-structure can explain this !

27. H.-W. Rix, Vatican 2003 Lensing Summary gravitational light deflection is important in many cosmological circumstances lensing has become a powerful cosmological tool confirmation of dark matter with relativistic (!) tracer conceptually independent measure of H 0 first demonstrated „passive“ evolution of the most masssive galaxies (not only in clusters) measures cosmological mass fluctuations (without dependence on galaxy distribution) galaxy halos are extended to > 200 kpc