Lecture 21 Cosmological Models

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Lecture 21Cosmological Models

• ASTR 340
• Fall 2006
• Dennis Papadopoulos

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Spectral Lines - Doppler
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Doppler Examples
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Doppler Examples
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Expansion Redshifts
z2 three times, z10, eleven times
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Expansion Redshifts
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Expansion - Example
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Current Record Redshift
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Hubbleology

• Hubble length DHc/H ,
• Hubble sphere Volume enclosed in Hubble sphere
  estimates the volume of the Universe that can be
  in our light-cone it is the limit of the
  observable Universe. Everything that could have
  affected us
• Every point has its own Hubble sphere
• Look-back time Time required for light to travel
  from emission to observation

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Gravitational Redshift
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Interpretation of Hubble law in terms of
relativity

• New way to look at redshifts observed by Hubble
• 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, as R(t) in the metric
  equation, also affects 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 local motions.

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Relation between z and R(t)

• Using our relativistic interpretation of cosmic
  redshifts, we write
• Redshift of a galaxy is defined by
• So, we have

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Hubble Law for nearby (zlt0.1) objects

• Thus
• where Hubbles constant is defined by
• But also, for comoving coordinates of two
  galaxies differing by space-time interval
• dR(t)?Dcomoving , have
• v Dcomoving ? ?R/?t(d/R)?(?R/?t)
• Hence v d? H for two galaxies with fixed
  comoving separation

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Peculiar velocities

• Of course, galaxies are not precisely at fixed
  comoving locations in space
• They have local random motions, called peculiar
  velocities
• e.g. motions of galaxies in local group
• This is the reason that observational Hubble law
  is not exact straight line but has scatter
• Since random velocities do not overall increase
  with comoving separation, but cosmological
  redshift does, it is necessary to measure fairly
  distant galaxies to determine the Hubble constant
  accurately

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Distance determinations further away

• In modern times, Cepheids in the Virgo galaxy
  cluster have been measured with Hubble Space
  Telescope (16 Mpc away)

Virgo cluster
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Tully-Fisher relation

• Tully-Fisher relationship (spiral galaxies)
• Correlation between
• width of particular emission line of hydrogen,
• Intrinsic luminosity of galaxy
• So, you can measure distance by
• Measuring width of line in spectrum
• Using TF relationship to work out intrinsic
  luminosity of galaxy
• Compare with observed brightness to determine
  distance
• Works out to about 200Mpc (then hydrogen line
  becomes too hard to measure)

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Hubble time

• Once the Hubble parameter has been determined
  accurately, it gives very useful information
  about age and size of the expanding Universe
• Recall Hubble parameter is ratio of rate of
  change of size of Universe to size of Universe
• If Universe were expanding at a constant rate, we
  would have ?R/?tconstant and R(t) t?(?R/?t)
  then would have H (?R/?t)/R1/t
• ie tH1/H would be age of Universe since Big Bang

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Modeling the Universe
Chapter 11
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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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• 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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Key Assumptions
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Riddles of Conventional Thinking
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Stability
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GR vs. Newtonian
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Newtonian Universe
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Expanding Sphere
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Fates of Expanding Universe
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Spherical Universe
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Friedman Universes
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Einsteins Greatest Blunder
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THE DYNAMICS OF THE UNIVERSE EINSTEINS MODEL

• Einsteins equations of GR

G describes the space-time curvature (including
its dependence with time) of Universe heres
where we plug in the RW geometries.
T describes the matter content of the Universe.
Heres where we tell the equations that the
Universe is homogeneous and isotropic.
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• Einstein plugged the three homogeneous/isotropic
  cases of the FRW metric formula into his
  equations of GR to see what would happen
• Einstein found
• That, for a static universe (R(t)constant), 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 this was before Hubble discovered expanding
  universe (more soon!) everybody thought that
  universe was static (neither expanding nor
  contracting).
• So instead, Einstein modified his GR 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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• Soon after, Hubble discovered that the universe
  was expanding!
• Einstein called the Cosmological Constant
  Greatest Blunder of My Life!
• .but very recent work suggests that he may have
  been right (more later!)

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Sum up Newtonian Universe