Gravitational Lensing as a Tool in Cosmology

1. 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/(c2r) = Rs/(2r) • 1911 Einstein: • finds the correct General Relativity answer  = 4GM/(c2r) = Rs/r • => differs by factor 2 from the Newtonian value H.-W. Rix, Vatican 2003

2. 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! H.-W. Rix, Vatican 2003

3. Lensing Basics • We consider the paths of light in the presence of masses (which curve space). • Assumptions: • Minkowski, or FRW, “smooth” space, with localizeddistortions • Local perturbations are weak • I.e. f << C2 andvsource,vlens,vobs << 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

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

5. From Blandford and Narayan 1986 View “onto” the sky Fermat’s principle in Gravitational Lensing (contd.) • Relativistic time dilation leads to an effective index of refraction neff=1+2|F|/c2 • Images are then formed, where is satisfied. • Images are formed in pairs should always expect to see odd number of images H.-W. Rix, Vatican 2003

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

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

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

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

10. Isothermal Sphere as a Lens • The total (stars + dark matter) mass profile is approximately isothermal, i.e. r~r-2  a simple, but applicable model for galaxies as lenses • because M(q)~q, the deflection is constant • Lens equation: q+ - = b +- qE with magnification m+ - = q+ - / b  image separation Dq is always2qE  image separationis direct measure of the enclosed mass H.-W. Rix, Vatican 2003

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

12. Lensed arcs are magnified pieces of the QSO host galaxy! Lens Modelling QSO 0957+561 Keeton et al 2002 • What can we learn from such lens systems • Mass distribution of lens • Structure of sourceNature’s telescope • Cosmological parameters, such as H0 • Procedure • Assume lens mass model • Map image back to source • Check match in source plane • Modify lens model  iterate lens galaxy H.-W. Rix, Vatican 2003

13. 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  H0 Note: Distance measurement not expansion velocity measurement  independent! H.-W. Rix, Vatican 2003

14. 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 • H0=67+-10 km/s/Mpc Note: time delay somewhat model dependent H.-W. Rix, Vatican 2003

15. 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 massivegalaxies H.-W. Rix, Vatican 2003

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

17. Cluster Mass Measurements E.g.: Cluster MS10from Ettori et al 2001 X-ray and lensing masses agree quite well! H.-W. Rix, Vatican 2003

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

19. 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 k and shear g (vector) H.-W. Rix, Vatican 2003

20. 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

21. 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

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

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

24. Projected Mass Overdensity Projected Radius Galaxy-Galaxy Lensing • 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

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

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

27. 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 H0 • 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 H.-W. Rix, Vatican 2003