1. 10/5/2004New Windows on the Universe Jan Kuijpers Part 1: Gravitation & relativityPart 1: Gravitation & relativity J.A. Peacock, Cosmological Physics, Chs. 1 & 2 Part 2: Classical CosmologyPart 2: Classical Cosmology Peacock, Chs 3 & 4

2. 10/5/2004New Windows on the Universe Part 2: Classical cosmology The isotropic universe (3)The isotropic universe (3) Gravitational lensing (4)Gravitational lensing (4)

3. 10/5/2004New Windows on the Universe The isotropic universe The RW metric (3.1)The RW metric (3.1) Dynamics of the expansion (3.2-3.3)Dynamics of the expansion (3.2-3.3) Observations (3.4)Observations (3.4)

4. 10/5/2004New Windows on the Universe Gravitational lensing Lense equation; lensing potential (4.1)Lense equation; lensing potential (4.1) Simple lenses (4.2)Simple lenses (4.2) Fermat’s principle (4.3)Fermat’s principle (4.3) Observations (4.4-4.6)Observations (4.4-4.6)

5. 10/5/2004New Windows on the Universe The RW metric (3.1) The isotropic universe Define fundamental observers: at rest in local matter distribution Global time coordinate t can be defined as proper time measured by these observers by these observers Choose radial coordinate so that either f=1 or g=r 2

6. 10/5/2004New Windows on the Universe The RW metric (3.1) Different definition of comoving distance r: Or dimensionless scale factor: Or isotropic form:

7. 10/5/2004New Windows on the Universe The RW metric (3.1) Or define conformal time:

8. 10/5/2004New Windows on the Universe Redshift Proper (small) separation of two fundamental observers: Hubble’s law Comoving distance between two fo’s is constant:

9. 10/5/2004New Windows on the Universe Dynamics of the expansion (3.2-3.3) GR required: - Birkhoff’s theorem - Integration constant - Integration constant Friedmann eqns: Use RW metric in field eqns (problem 3.1): Newton.: 1. Energy eqn. Take time derivative + energy conservation

10. 10/5/2004New Windows on the Universe Flatness problem Flatness problem Matter radiation equality: Matter radiation equality: Recombination: Recombination: 1+z rec =1000 Matter dominated and flat: Matter dominated and flat: Radiation dominated and flat: Radiation dominated and flat: Vacuum energy follows from energy conservation) Vacuum energy (p=-  c 2 follows from energy conservation):

11. 10/5/2004New Windows on the Universe Observations (3.4) Luminosity distance: the apparent distance assuming inverse square law for light intensity reduction -Luminosity : power output/4  -Luminosity L : power output/4  -Radiation flux density : energy received per unit area per sec -Radiation flux density S: energy received per unit area per sec Redshift for photon energy and one for rate Angular-diameter distance: the apparent distance based on observed diameter assuming euclidean universe

12. 10/5/2004New Windows on the Universe Lensing equation; lensing potential (4.1) Gravitational lensing Relativistic particles in weak fields (eq. 2.24): Bend angle (use angular diameter distances): Approximation: geometrically thin lenses

13. 10/5/2004New Windows on the Universe Gravitational lenses are flawed!!!

14. 10/5/2004New Windows on the Universe Gravitational imaging

15. 10/5/2004New Windows on the Universe Lensing equation DLDL D LS DSDS Flux density from image is:

16. 10/5/2004New Windows on the Universe Lensing potential Notation: - potential!

17. 10/5/2004New Windows on the Universe Multiple images Simple lenses (4.2) D LS DLDL Circularly symmetric surface mass density:

18. 10/5/2004New Windows on the Universe Einstein ring SO L r

19. 10/5/2004New Windows on the Universe Typical numbers Einstein Radius point mass: ER isothermal sphere: Critical surface density:

20. 10/5/2004New Windows on the Universe DLDL D LS DSDS b Time delays Time lags between multiple images because of: 1.Path length difference: 2. Reduced coordinate speed of light (static weak fields):

21. 10/5/2004New Windows on the Universe Fermat’s principle (4.3) Images form along paths where the time delay is stationary Note: differentiation wrt  I recovers lens equation. Example: from a to d: introduction of increasing mass (increasing -  ) leads to extra Stationary points (minima, Maxima, saddle points in  )

22. 10/5/2004New Windows on the Universe Caustics and catastrophe theory

23. 10/5/2004New Windows on the Universe Lens model for flattened galaxy at two different relative distances. a: density contours c: caustics in image plane b: time surface contours d: dual caustics in source plane

24. 10/5/2004New Windows on the Universe Observations (4.4-4.6) Light deflection around the Sun The Sun 1.75”

25. 10/5/2004New Windows on the Universe Newton/Soldner versus Einstein

26. 10/5/2004New Windows on the Universe

27. 10/5/2004New Windows on the Universe Total eclipse 21 september 1922 Western Australia, 92 stars (dots are reference positions, lines displacements, enlarged!)

28. 10/5/2004New Windows on the Universe Strong lensing

29. 10/5/2004New Windows on the Universe

30. 10/5/2004New Windows on the Universe Modelling

31. 10/5/2004New Windows on the Universe

32. 10/5/2004New Windows on the Universe

33. 10/5/2004New Windows on the Universe

34. 10/5/2004New Windows on the Universe

35. 10/5/2004New Windows on the Universe Robert J. Nemiroff 1993: Sky as seen past a compact star, 1/3 bigger than its Schwarzschild radius, and at a distance of 10 Schwarzschild radii. The star has a terrestrial surface topography

36. 10/5/2004New Windows on the Universe Orion Orion Sirius Sirius

37. 10/5/2004New Windows on the Universe