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

H.-W. Rix, Vatican 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 – first lens!

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

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

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

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

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

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

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

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

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

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 Keeton et al 2002  lens galaxy

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 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!

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 = km/s/Mpc Note: time delay somewhat model dependent

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 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

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

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!

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

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)

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!

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

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

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

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)

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

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 !

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