1. PREDRAG JOVANOVIĆ AND LUKA Č. POPOVIĆ ASTRONOMICAL OBSERVATORY BELGRADE, SERBIA Gravitational Lensing Statistics and Cosmology

2. Outline Observational cosmology: basics and parameters Cosmological experiments: 1. Cosmic Microwave Background Radiation (CMBR) 2. Type Ia supernovae 3. Gravitational lensing  Strong: detection of distant galaxies  Weak: detection of dark matter  Time delay: determination of H 0  Statistics: constraining Ω 0 and Ω Λ  Problems with gravitational lensing statistics Conclusions

3. Cosmology basics The current models of cosmology are based on the field equations of general relativity: Friedmann-Lemaître-Robertson-Walker (FLRW) metric: a solution of the Einstein field equations in the case of a simply connected, homogeneous, isotropic expanding or contracting universe: r, ϕ, ϑ - comoving polar coordinates k - the scalar curvature of the 3-space: k = 0, > 0, or < 0 corresponds to flat, closed, or open universe a(t) - the dimensionless scale factor of the universe ΛCDM model uses the FLRW metric, the Friedmann equations and the cosmological equation of state to describe the universe

4. Cosmological parameters H - the Hubble constant ρ - the mass density of the universe Λ - the cosmological constant k - the curvature of space a - the expansion factor of universe dimensionless density parameters: where the subscript “0” indicate the quantities which in general evolve with time and which are referring to the present epoch several observational techniques are used for their estimation

5. Wilkinson Microwave Anisotropy Probe (WMAP) The "angular spectrum" of the fluctuations in the WMAP full-sky map, showing the relative brightness of the "spots" in the map vs. the size of the spots. The shape of this curve contain a wealth of information about the history of the universe

6. Supernova Cosmology Project Type Ia supernovae: the standard candles Intrinsic luminosity is known Apparent luminosity can be measured The ratio of above two luminosities can provide the luminosity-distance (d L ) of a SN The red shift z can be measured independently from spectroscopy Using d L (z) or equivalently the magnitude(z) one can draw a Hubble diagram

7. Constraining the cosmological parameters Riess et al. 2004, ApJ, 607, 665 Tonry et al. 2003, ApJ, 594, 1

8. Content of the Universe

9. Gravitational lensing

10. Einstein Ring Radius of a gravitational lens

11. QSO 2237+030 (z=1.695), also known as “Einstein cross” and lensing galaxy ZW2237+030 (z=0.0394) RXJ1131-1231 PG 1115+080 Examples:

12. Strong lensing: detection of distant galaxies The orange arc: an elliptical galaxy at z=0.7, the blue arcs: star forming galaxies at z= 1 - 2.5 the red arc and the red dot: the farthest known galaxy at z~7 (13 billion ly away, i.e. only 750 million years after the big bang

13. Weak lensing: detection of dark matter unlensed lensed

14. Distribution of dark matter

15. The Hubble constant from gravitational lens time delays Kochanek & Schechter, 2003, astro-ph/0306040

16. Courbin, 2003, astro-ph/0304497 HST Key Project: determination of the H 0 by the systematic observations of Cepheid variable stars in several galaxies using HST

17. Gravitational lensing statistics More details about history and basics in the book: P. Schneider, C. Kochanek and J. Wambsganss, 2006, “Gravitational Lensing: Strong, Weak and Micro”, Saas-Fee Advanced Courses, Springer Berlin Heidelberg (http://www.springerlink.com/content/n37347/)http://www.springerlink.com/content/n37347/ Optical depth for gravitational lensing, i.e. the probability to observe such effects (Turner et al. 1984, ApJ, 284, 1; Turner, 1990, ApJ, 365, L43): where z S and z L are the source and lens redshifts, σ is lens velocity dispersion,  (σ; z L ) is the velocity function, A is the cross section for multiple imaging, B is the magnification bias, dV is the differential comoving volume element The Current State: lens statistics constraints on   and  0 are in good agreement with results from Type Ia supernovae for a spatially flat universe:   = 0.72 - 0.78 (Mitchell et al. 2005, ApJ, 622, 81)

18. Likelihood contours at the 68%, 90%, 95%, and 99% confidence levels. The dotted line marks spatially flat cosmologies The separation distribution of the 12 CLASS lenses Mitchell et al. 2005, ApJ, 622, 81 Differential (thick) and cumulative (thin) probability along the line of spatially flat cosmologies

19. Gravitational macrolensing optical depth The effective optical depth is related to the number N GL (z) of multiply imaged quasars within a sample of N QSO (z) quasars with redshifts z by: Zakharov, Popović and Jovanović, 2004, A&A, 881

20. Distribution of all QSOs and lensed QSOs in Veron & Veron Catalogue Veron-Cetty & Veron, 2006, A&A, 455, 773: a sample of 85221 (N QSO ) quasars among which 69 (N GL ) are gravitationally lensed

21. The ratio of lensed to total number of quasars and optical depth for three different flat cosmological models as a function of quasar redshift

22. Optical depth of cosmologically distributed gravitational microlenses (Zakharov, Popović and Jovanović, 2004, A&A, 881)

23. Optical depth of cosmologically distributed gravitational microlenses for three different values of  L

25. Problems with gravitational lensing statistics Small number of observed gravitational lenses (~100) is insufficient for reliable statistics. Solution not later than 2015: LSST, SNAP, SKA and JWST projects will drastically increase the number of detected gravitational lenses Large Synoptic Survey Telescope (LSST): 2013 SuperNova/Acceleration Probe (SNAP): 2013 Square Kilometre Array (SKA): 2015 James Webb Space Telescope (JWST): 2013 Extinction by dust in the lens galaxies leads to artificially low number of observed lenses Galaxy evolution: decrease of lensing population for higher redshifts would lower the number of observed lenses Ellipticity and clustering: mass distributions of lenses is not circularly symmetric Cosmology

26. Conclusions We demonstrated constraining the cosmological parameters by gravitational lens statistics on a sample of lensed quasars from Veron & Veron catalogue of quasars and active nuclei Obtained results are in satisfactory agreement with those obtained from CLASS and SDSS surveys (Mitchell et al. 2005, ApJ, 622, 81) Optical depth of cosmologically distributed gravitational microlenses also depends on assumed cosmological model (Zakharov, Popović and Jovanović, 2004, A&A, 881)