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

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 

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

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

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

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”

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

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) 

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) 

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”

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)

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

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

Today’s big question: Slope = Hubble constant

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