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Lecture 1: Preliminaries and Cosmology in a Homogeneous Universe

Lecture 1: Preliminaries and Cosmology in a Homogeneous Universe. Basics of Astronomical Objects. Stars: 0.1 – 100 M Sun M Sun ~2 x 10 33 g R Sun ~ 7 x 10 10 cm Luminosity (bolometric) ~4 x 10 33 ergs/sec Lifetime: 10 6 yrs to >10 10 yrs Galaxies:

ishmael-olson
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Lecture 1: Preliminaries and Cosmology in a Homogeneous Universe

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  1. Lecture 1: Preliminaries andCosmology in a Homogeneous Universe

  2. Basics of Astronomical Objects • Stars: 0.1 – 100 MSun • MSun~2 x 1033 g • RSun ~ 7 x 1010 cm • Luminosity (bolometric) ~4 x 1033 ergs/sec • Lifetime: 106 yrs to >1010 yrs • Galaxies: • (Stellar) Masses: ~107MSun to 1011MSun • Sizes: R0.5 ~ 0.5–10 kpc (1pc=3x1018 cm) • Characteristic separation 1-5 Mpc (between 1010MSun galaxies)

  3. Basics (contd.) • “Universe”: • Current expansion rate 70+-5 km/s/Mpc (“Hubble constant”) In real units: H0~ (14 Gyrs )-1  rough age of Universe

  4. Averaged over sufficiently large scales, the universe is nearly homogeneous and isotropic(=cosmological principle) • The universe, i.e. space itself, is expanding so that the distance D between any pairs of widely separated points increases as dD/dt~D (=Hubble law) • ? the universe expanded from a very dense, hot initial state(=big bang) • The expansion of the universe is determined by its mass/energy content and the laws of General Relativity • On “small scales” (<10-100 Mpc) a great deal of structure has formed through gravitational self-organization. Basic pillars of our cosmological picture (i.e. we are starting with the answer first)

  5. The Cosmological Principle the Universe is • homogeneous (uniform density) • and isotropic (looks the same in all directions) (Hubble’s Law is a natural outcome in a homogeneous, isotropic, expanding universe)

  6. (isotropy implies homogeneity but not vice versa)

  7. the Universe is clumpy on small scales but smooth on large scales

  8. q R what is a metric? Euclidean (flat) space: Cartesian 2-D polar

  9. P1 q in curved space r f P2 R = radius of sphere r = geodesic distance

  10. alternatively, we can define the coordinate where x is an ‘angular size distance’ then, curvature

  11. metric in curved, 3D space (not easy to draw!): in terms of the geodesic distance in terms of the angular size distance

  12. now adding the fourth dimension (time): Minkowski metric

  13. The Robertson-Walker Metric define a function a(t) that describes the dynamics of the expansion: a: scale factor r: comoving coordinate radius of curvature

  14. or, in terms of the angular size distance variable: note: we can define coordinates such that a(t0)=1 or such that k=-1, 0, 1 for negative, flat, or positive curvature but not both simultaneously.

  15. General Relativity says: • mass tells space-time how to curve • the curvature of space-time tells mass how to move • therefore, the amount of mass in the Universe also determines the geometry of space-time

  16. c2r p p p Tmn = Einstein’s Field Equations metric in tensor form Rmn - ½ R gmn + Lgmn = 8pGTmn Cosmological Constant Energy-momentum Tensor Einstein Tensor (curvature) where T00 = energy density T12 = x-component of current of momentum in x direction, etc. e.g., for a perfect fluid:

  17. The Friedmann equations

  18. The equation of state p = wr relates pressure and density matter w = 0 radiation w = 1/3 vacuum w = -1

  19. scale factor: a(te)ne = a(to)no a = 1/(1+z) cosmological redshift z = (lo-le)/ le

  20. back to Hubble... proper separation of two fundamental observers is a(t) dr H0 = H(a0) photons travel on null geodesics of zero proper time, so comoving distance

  21. critical density and W for k=0

  22. components of the Universe matter radiation vacuum energy

  23. The Friedmann equation recast

  24. solutions of the Friedmann eqn W=1: a(t) = (t/t0)2/3  t0 = 2/3 1/H0 W<1: a(t) = ½ [W0 / (1-W0)] (cosh q -1) t = 1/(2H0) [W0 / (1-W0)]3/2 (sinh q - q)

  25. solutions to the Friedmann equation scale factor a(t)

  26. W < 1 W = 1 W > 1

  27. the matter density W, the geometry, and the fate of the Universe are all interconnected

  28. distances in cosmology • comoving coordinate r • appears in metric; not directly observable • proper distance l = r a(t) • luminosity distance dL=[L/4pf]1/2 • relates flux and luminosity • angular diameter distance dA=D/q • relates angular and physical size (diameter)

  29. open flat EdS

  30. open flat EdS

  31. open flat EdS

  32. flat open EdS

  33. Summary • the Robertson-Walker metric is the most general for a homogeneous, isotropic universe containing matter • the Friedmann eqn describes the relationship between the expansion, the geometry, and the energy density of the universe • in cosmology, not all distances are equal.

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