Lecture 14 : Cosmological Models I

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Lecture 14 Cosmological Models I

• The basic Cosmological Principles
• The geometry of the Universe
• The scale factor R, curvature constant k
• Einsteins cosmology
• FRW models
• For background, read chapters 10 11 from
  textbook.

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I BASIC COSMOLOGICAL ASSUMPTIONS

• Germany 1915
• Einstein just completed theory of GR
• Explains anomalous orbit of Mercury perfectly
• Schwarzschild is working on black holes etc.
• Einstein turns his attention to modeling the
  universe as a whole
• How to proceed its a horribly complex problem

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How to make progress

• Proceed by ignoring details
• Imagine that all matter in universe is smoothed
  out
• i.e., ignore details like stars and galaxies, but
  deal with a smooth distribution of matter
• Then make the following assumptions
• Universe is homogeneous every place in the
  universe has the same conditions as every other
  place, on average.
• Universe is isotropic there is no preferred
  direction in the universe, on average.

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Observational evidence for homogeneity and
isotropy

• Lets look into space see how matter is
  distributed on large scales.
• Redshift surveys
• Make 3-d map of galaxy positions
• Use redshift Hubbles law to determine distance

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Each point is a bright galaxy
CfA redshift survey
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Las Campanas Redshift survey
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• There is clearly large-scale structure
• Filaments, clumps
• Voids and bubbles
• But, homogeneous on very large-scales.
• So, we have the
• The Generalized Copernican Principle there are
  no special points in space within the Universe.
  The Universe has no center!
• These ideas are collectively called the
  Cosmological Principles.

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II POSSIBLE GEOMETRIES FOR THE UNIVERSE

• The Cosmological Principles constrain the
  possible geometries for the space-time that
  describes Universe on large scales.
• The problem at hand - to find curved 4-d
  space-times which are both homogeneous and
  isotropic
• Solution to this mathematical problem is the
  Friedmann-Robertson-Walker (FRW) metric.

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The FRW metric

• Introduce the curvature constant, k
• Three possible cases
• Spherical spaces (closed k1)

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• Flat spaces (open k0)
• Hyperbolic spaces (open k-1)

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The scale factor, R.

• Scale factor, R, is a central concept!
• R tells you how big the space is
• Allows you to talk about expansion and
  contraction of the universe (even if Universe is
  infinite).
• Simplest example is k1 case (sphere)
• Scale factor is just the radius of the sphere

R0.5
R1
R2
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• What about k-1 (hyperbolic) universe?
• Scale factor gives radius of curvature
• For k0 universe, there is no curvature shape is
  unchanged as universe changes its scale
  (stretching a flat rubber sheet)

R1
R2
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III THE DYNAMICS OF THE UNIVERSE EINSTEINS
MODEL

• Back to Einsteins equations of GR
• For now, ignore cosmological constant

T describes the matter content of the Universe.
Heres where we tell the equations that the
Universe is homogeneous and isotropic.
G describes the curvature (including its
dependence with time) of Universe heres where
we plug in the FRW geometries.
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• Einstein plugged the three homogeneous/isotropic
  cases into his equations of GR to see what would
  happen
• but this was before Hubble discovered expanding
  universe everybody thought that universe was
  static (neither expanding nor contracting).
• Einstein found
• That, for a static universe, only the spherical
  case worked as a solution to his equations
• If the sphere started off static, it would
  rapidly start collapsing (since gravity attracts)
• The only way to prevent collapse was for the
  universe to start off expanding there would then
  be a phase of expansion followed by a phase of
  collapse

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• So Einstein could have used this to predict that
  the universe must be either expanding or
  contracting!
• But he took a wrong turn
• Einstein modified his equations
• Essentially added a repulsive component of
  gravity
• New term called Cosmological Constant
• Could make his spherical universe remain static
• BUT, it was unstable a fine balance of opposing
  forces. Slightest push could make it expand
  violently or collapse horribly.

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• Then Hubble discovered that the universe was
  expanding!
• Einstein called the Cosmological Constant
  Greatest Blunder of My Life!

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IV THE STANDARD COSMOLOGICAL MODEL

• In general Einsteins equation relate geometry to
  dynamics
• Turns out that there are three possibilities

From web site of the University of Oregon
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Important features of standard models

• All models begin with R0 at a finite time in the
  past
• This time is known as the BIG BANG
• Space-time curvature is infinite at the big bang
• Space and time come into existence at this
  moment there is no time before the big bang!
• The big-bang happens everywhere in space not at
  a point!

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• There is a connection between the geometry and
  the dynamics
• Closed (k1) universes re-collapse
• Open (k-1) universes expand forever
• Flat (k0) universe expand forever (but only
  just they almost grind to a halt).

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V CONNECTING STANDARD MODELS AND HUBBLES LAW

• New way to look at redshift observed by Hubble
  and Slipher
• Redshift is not due to velocity of galaxies
• Galaxies are (approximately) stationary in space
  
• Galaxies get further apart because the space
  between them is physically expanding!
• The expansion of space also effects the
  wavelength of light as space expands, the
  wavelength expands and so there is a redshift.
• So, cosmological redshift is due to cosmological
  expansion of wavelength of light, not the regular
  Doppler shift from galaxy motions.

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Relation between z and R(t)

• Redshift of a galaxy given by
• Using our new view of redshift, we write
• So, we have

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Co-moving coordinates.

• Lets formalize this Consider a set of
  co-ordinates that expand with the space. We call
  these co-moving coordinates.

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• If a galaxy remains at rest once the overall
  expansion has been accounted for, then it has
  fixed co-moving coordinates.
• Consider two galaxies that have fixed co-moving
  coordinates.
• Lets define a co-moving distance D
• Then, the real (proper) distance between the
  galaxies is proper distance R(t) D

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• So.. Hubbles constant given by