1. Expanding Universes Intro Cosmology Short Course Lecture 2 Paul Stankus, ORNL

2. Henrietta Leavitt American Distances via variable stars Edwin Hubble American Galaxies outside Milky Way The original Hubble Diagram “A Relation Between Distance and Radial Velocity Among Extra- Galactic Nebulae” E.Hubble (1929) Distance  Velocity 

3. US Slightly Earlier As seen from our position: Same pattern seen by all observers! US Now As seen from another position: US

4. Freedman, et al. Astrophys. J. 553, 47 (2001) W. Freedman Canadian Modern Hubble constant (2001) 1929: H 0 ~500 km/sec/Mpc 2001: H 0 = 72  7 km/sec/Mpc Hubble relation: v Recession  H 0 d Modern Hubble Diagram Original Hubble diagram

5. ? H(Earlier)  H(Now) H(Earlier)  H 0 v Recession  H 0 d 1/H 0 ~ 10 10 year ~ Age of the Universe? Photons t Us Galaxies 1/H 0 x Now t ? 1/H Earlier x Earlier

6. A. Friedmann Russian G. LeMaitre Belgian Evolution of homogeneous, non- static (expanding) universes “Friedmann models” (1922, 1927) Friedmann-LeMatire Cosmologies 1.Isotropy Same in all directions 2.Homogeneity Same at all locations “Cosmological Principle” “Copernican Principle”

7. Warning: No Newtonian Solution (Get used to it, get over it) Test Particle Spherical Shell Test Particle Uniform Sphere Spherical Shell F, a  0F, a  0

8. H.P. Robertson American A.G. Walker British Formalized most general form of isotropic and homogeneous universe in GR “Robertson-Walker metric” (1935-6) Space-like Hyper-surfaces Each at constant t Energy density, spatial curvature, and pressure are all uniform across each surface Free-fall Paths at constant  Surfaces t are simultaneous to these observers Choose t such that d  dt along these paths Spatial  Coordinate Systems Distances within each space-like hyper-surface all obey d  a(t) 

9. Robertson-Walker Metric Minkowski Metric 1.The  coordinate system is uniform (not necessarily Euclidean) 2.  has units of length 3. a is dimensionless 4.By convention a(now)  a 0  1 5.Physical distance for  on surface t is d  a(t) 

10. Three basic (Euclidean) solutions for a(t): 1. Relativistic gas, “radiation dominated” P/  = 1/3  a -4 a(t)  t 1/2 2. Non-relativistic gas, “matter dominated” P/  = 0  a -3 a(t)  t 2/3 3. “Cosmological-constant-dominated” or “vacuum-energy-dominated” P/  = -1  constant a(t)  e Ht “de Sitter space”

11. Us Galaxies “at rest in the Hubble flow” t t  now  Galaxies Hubble “Constant” Robertson-Walker Coordinates Galaxies t  earlier  Current separation d (now)  a(now)   a 0  Earlier separation d (earlier)  a(earlier)  d (now)

12. Photons Us t  Photons Robertson-Walker Coordinates Photons t  now Big Bang Slope = 0 a(t) = 0 Observable Size Age

13. Robertson-Walker Coordinates Cosmological Red-Shift t  t1t1 t2t2 Photons A photon’s period grows  a(t) Its coordinate wavelength  is constant; its physical wavelength a(t)  grows  a(t) (t 2 )/ (t 1 ) = a(t 2 )/a(t 1 )  1+z Cosmological Red Shift Period Wavelength

14. Today’s big question: Slope = Hubble constant

15. Points to take home Hubble expansion: recession velocity  distance away Uniform expansion pattern Describe in GR by Robertson-Walker metric Galaxies “at rest in the flow,” but physically separating Photon paths simple, derive cosmological red-shift