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Gravitational Lensing as a Tool in Cosmology

Explore the historical journey of gravitational lensing in cosmology, from Newton's insights to Einstein's General Relativity and modern lens modeling techniques for galaxy masses. Discover the fundamentals of light deflection and image formation in lensing phenomena.

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Gravitational Lensing as a Tool in Cosmology

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  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

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