1. April 3, 2005 The lens redshift distribution – Constraints on galaxy mass evolution Eran Ofek, Hans-Walter Rix, Dan Maoz (2003)

2. April 3, 2005 Talk layout Basics of gravitational lensing The lens redshift distribution test Sample selection Constraints on galaxy mass evolution The maximum likelihood and bias

3. April 3, 2005 Gravitational Lensing DsDs D ls DlDl

4. April 3, 2005 Gravitational Lensing

5. April 3, 2005 Gravitational Lensing

6. April 3, 2005 Lens Redshift Distribution Based on observations: θ depends on lens mass (= velocity dispersion  ) + f/p Number density  velocity dispersion (=mass) ** n*n* Evolution? n * = n * (redshift)  * =  * (redshift) Probability for Lensing P(z l | z s,  ) ~ Number density of Lenses x Given Lens: probability of θ x Cosmology How much “distance” in redshift interval

7. April 3, 2005 Lens Redshift Distribution - Example

8. April 3, 2005 Selecting a sample Doesn’t depend on angular selection completeness Image separation is prior Kochanek (1996): truncated the redshift probability distribution beyond the redshift at which the lens galaxy become too faint to have its redshift measured Missing redshift information? pr

9. April 3, 2005 The Sample Source redshift cut Sample I N=14 Sample II N=11 More than 80 gravitational lenses known Ignoring lenses without redshift information Bias We need lenses with complete redshift information

10. April 3, 2005 Probability vs. Lenses JK

11. April 3, 2005 Galaxy Mass Evolution L definitions cos U/P/T

12. April 3, 2005 Galaxy Mass Evolution Not probable rejected

13. April 3, 2005 Cosmology No evolution Flat Universe

14. April 3, 2005 Jackknife - bias and outliners Bias: Given a sample S 1,....,S n : calculate statistic T Drop point number k (=S k ). Calculate T(S 1,...,S k-1,S k+1,...,S n ) = T i≠k Quenouille-Tukey jackknife bias:

15. April 3, 2005 Jackknife - bias and outliners Bias:

16. April 3, 2005 Summary Assuming “standard cosmology” Sensitive to galaxy mass evolution Assuming no evolution

17. April 3, 2005 End

18. April 3, 2005 Lens Redshift Distribution Number density of lenses = galaxy mass function Proper distance interval Lensing cross section + f/p Schechter function Faber-Jackson relation

19. April 3, 2005 Sensitivity definitions FJ

20. April 3, 2005 Future Work Obtain additional lens redshift Maoz, Rix, & Kochanek (@ VLT) Use velocity dispersion function e.g., Sheth et al. (2003) Larger sample Higher redshift (z~1.5)

21. April 3, 2005 Spherical Isothermal Sphere SIS assumption is consistent with data: - Galaxy dynamics (Rix et al. 1997) - X-ray emission from ellipticals (Fabbiano et al. 1989) - Lensing (Treu & Koopmans 2002) SIS lenses Einstein radius = image separation For SIS the Einstein radius:

22. April 3, 2005 Previous Work Kochanek 1992 Helbig & Kayser 1996 Kochanek 1996 Weak constraints on cosmology N=4 N=6 N=8 Sample size

23. April 3, 2005 Lens Redshift Distribution SpiralS0Elliptical Parameters : 2dF - Madgwick et al. (2002)

24. April 3, 2005 Problems Lenses discovered based on lens-properties Missing redshifts Galaxy evolution Clusters Mass profile of galaxies Q2237+0305 Photometric Redshift - FBQ0951+2635 Martel et al. (2002) Keeton et al. (2000) Effect on separation Kochanek (1996) - small effect

25. April 3, 2005 Galaxy Evolution Van Dokkum et al. (1998)0.83 Keeton, Kochanek & Falco (1998)~1 Lin et al. (1999)0.55 van Dokkum et al. (2001)0.55 Cohen (2002)~1 Rusin et al. (2002)~1 Im et al. (2002)~1

26. April 3, 2005 Scatter of the Faber-Jackson unperturbed 20% ln-normal scatter 40% ln-normal scatter

27. April 3, 2005 Sensitivity - Cosmology Flat Universe

28. April 3, 2005

29. Gravitational Lensing DsDs D ls DlDl

30. April 3, 2005 Gravitational Lensing

31. April 3, 2005 Gravitational Lensing DsDs D ls DlDl