1. Meaning of deceleration parameter flat model q 0 tells you if the Universe is accelerating in its global expansion rate, or decelerating. Flat, matter dominated q0=1/2 Empty Universe q0=0 Flat, radiation dominated q0=1 Flat, Cosmological Constant dominated q0=-1

2. Equation of State Revisited

3. Distance as a function of redshift

4. Exact Distance Solutions for open/closed & mixed models

5. Integrating functions Use trapezoid rule f(x) x1 x2 x1x1+  x f(x1)f(x1+  x) Area   x(f(x1+  x)+ f(x1))/2

6. Trial numerical integration

7. Angular Size Distance How big does an object look as a function of redshift Proper Motion Distance Apparent Angular velocity as a function of redshift Luminosity Distance How bright does an object appear as a function of redshift

8. Cosmic Chronology How old is an object as a function of redshift Volume Test How does the Volume per dz change as a function of z Structure Test How does structure evolve relative to a model Density Test Measure local density of Matter

9. Luminosity Distance – GR + Homogenous Isotropic Universe for a monochromatic source (defined as inverse-square law) the flux an observer sees of an object at redshift z Brightness of object depends exclusively on what is in the Universe How much and its equation of state.

10. The Deceleration Parameter Valid for all models Mattig (1958) provides exact solution for matter only Universes

11. Distance Modulus Fractional distance change  ½(mag change) e.g. 0.1mag difference is 5% distance difference

12. Angular Size Distance

14. Attempts to fit/understand models: 1920-1995 Number counts of Galaxies – Hubble,Yoshii/Peterson Angular Size Distances - distant radio cores Kellerman Loitering Universes with z=2 Quasars Luminosity Distance with Brightest Cluster Galaxies

15. Volume Effects At low z, N  z 3, for any non-diverging luminosity function, Hubble observed this to be true, demonstrating that Galaxies uniformly fill the local Universe. He and Humason were unable to reach conclusions on Geometry.

17. Angular Size Distance with Compact Radio Sources Kellerman (1993)

18. Stepanas & Saha 1995 Result not that constraining

19. Excess z~2 QSOs: Loitering Universe Petrosian, V., Saltpeter, E.E. & Szekeres, P. 1967

20. Brightest Cluster Galaxies Sandage, Humason & Mayhall 1956 Baum 1957 Peach 1970 Deceleration q 0 >1 But Tinsley 1976 showed Evolution dominates Cosmology

21. Methods for Measuring Extra- Galactic Distances Brightest Cluster Galaxies Cepheids Fundamental Plane (D n -  /Faber Jackson) Lensing Delay Planetary Nebulae Tully-Fisher Sunyaev-Zeldovich Surface Brightness Fluctuations Supernovae Ia Supernovae II