Lecture 1: Preliminaries and Cosmology in a Homogeneous Universe

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Lecture 1 Preliminaries andCosmology in a
Homogeneous Universe
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Basics of Astronomical Objects

• Stars 0.1 100 MSun
• MSun2 x 1033 g
• RSun 7 x 1010 cm
• Luminosity (bolometric) 4 x 1033 ergs/sec
• Lifetime 106 yrs to gt1010 yrs
• Galaxies
• (Stellar) Masses 107MSun to 1011MSun
• Sizes R0.5 0.510 kpc (1pc3x1018 cm)
• Characteristic separation 1-5 Mpc
• (between 1010 MSun galaxies)

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Basics (contd.)

• Universe
• Current expansion rate 70-5 km/s/Mpc
• (Hubble constant)
• In real units H0 (14 Gyrs )-1 ? rough age of
  Universe

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• Basic pillars of our cosmological picture
• (i.e. we are starting with the answer first)

• 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/dtD (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 (lt10-100 Mpc) a great deal of
  structure has formed through gravitational
  self-organization.

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The Cosmological Principle

• the Universe is
• homogeneous (uniform density)
• and isotropic (looks the same in all directions)

(Hubbles Law is a natural outcome in a
homogeneous, isotropic, expanding universe)
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(isotropy implies homogeneity but not vice versa)
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the Universe is clumpy on small scales but smooth
on large scales
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what is a metric?
Euclidean (flat) space
Cartesian
2-D polar
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P1
q
in curved space
r
f
P2
R radius of sphere r geodesic distance
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alternatively, we can define the coordinate
where x is an angular size distance
then,
curvature
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metric in curved, 3D space (not easy to draw!)
in terms of the geodesic distance
in terms of the angular size distance
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now adding the fourth dimension (time)
Minkowski metric
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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
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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.
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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

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Einsteins 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
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The Friedmann equations
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The equation of state
p wr
relates pressure and density
matter w 0 radiation w
1/3 vacuum w -1
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scale factor a(te)ne a(to)no
a 1/(1z)
cosmological redshift z (lo-le)/ le
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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
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critical density and W
for k0
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components of the Universe
matter
radiation
vacuum energy
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The Friedmann equation recast
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solutions of the Friedmann eqn
W1 a(t) (t/t0)2/3 ? t0 2/3 1/H0
Wlt1 a(t) ½ W0 / (1-W0) (cosh q -1) t
1/(2H0) W0 / (1-W0)3/2 (sinh q - q)
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solutions to the Friedmann equation
scale factor a(t)
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W lt 1
W 1
W gt 1
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the matter density W, the geometry, and the fate
of the Universe are all interconnected
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distances in cosmology

• comoving coordinate r
• appears in metric not directly observable
• proper distance l r a(t)
• luminosity distance dLL/4pf1/2
• relates flux and luminosity
• angular diameter distance dAD/q
• relates angular and physical size (diameter)

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open
flat
EdS
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open
flat
EdS
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open
flat
EdS
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flat
open
EdS
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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.