Energetic Tail of Photon Bath

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Energetic Tail of Photon Bath
hv
hardest photons
baryons
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Evolution of Sound Speed
Expand a box of fluid
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Radiation
Matter
Matter number density
Random motion energy Non-Relativistic IDEAL GAS

• Show C2s c2/3 /(1Q) , Q (3 ?m) /(4 ?r) , ?
  Cs drops
• from c/sqrt(3) at radiation-dominated era
• to c/sqrt(5.25) at matter-radiation equality

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hv
hv
Compton-scatter
Keep electrons hot Te Tr until redshift z
After decoupling (zlt500), Cs 6 (1z) m/s
because
T?
Te
R
Until reionization z 10 by stars quasars
Cs 6 (1z) m/s
dP
dP
dX
dX
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• Growth of Density Perturbations and peculiar
  velocity

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Peculiar Motion

• The motion of a galaxy has two parts

Proper length vector
Uniform expansion vo
Peculiar motion ?v
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Damping of peculiar motion (in the absence of
overdensity)

• Generally peculiar velocity drops with expansion.

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Non-linear Collapse of an Overdense Sphere

• An overdense sphere is a very useful non linear
  model as it behaves in exactly the same way as a
  closed sub-universe.
• The density perturbations need not be a uniform
  sphere any spherically symmetric perturbation
  will clearly evolve at a given radius in the same
  way as a uniform sphere containing the same
  amount of mass.

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R, R1
log?
Rmax
t-2
Rmax/2 virialize
t
Background density changes this way
logt
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Gradual Growth of perturbation
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Equations governing Fluid Motion
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• Let
• We define the Fractional Density Perturbation

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• Motion driven by gravity
• due to an overdensity
• Gravity and overdensity by Poissons equation
• Continuity equation
• Peculiar motion and peculiar gravity both scale
  with d and are in the same direction.

The over density will rise if there is an inflow
of matter
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the equation for linear growth

• At high zgtgt1
• matter domination
• In the equation

Gravity has the tendency to make the density
perturbation grow exponentially.
Pressure makes it oscillate
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Nearly Empty Pressure-less Universe
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The Jeans Instability

• Case 1- no expansion
• Assume the density contrast ? has a wave-like
  form
• Assume no expansion
• ? the dispersion relation

Pressure support
gravity
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• At the (proper) JEANS LENGTH scale we switch from
  
• standing sound waves for shorter wavelengths to
• the exponential growth of perturbations for long
  wavelength modes
• ?lt?J, ?2gt0 ? oscillation of the perturbation.
• ???J, ?2?0?exponential growth/decay
• 
  

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• Timescale
• Application Collapse of clouds, star formation.

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Jeans Instability

• Case 2 on very large scale ?gtgt?J of Expanding
  universe
• Neglect Pressure (restoring force) term

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• Einstein de Sitter
• Generally

?M1
log?
Log R/R0
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Case III Relativistic Fluid

• equation governing the growth of perturbations
  being

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Jeans Mass Depends on the Species of the Fluid
that dominates

• If Photon dominates
• If DarkMatt dominates decoupled from photon

cstdistance travelled since big bang
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• Jeans Mass past and now

Flattens out at time of equality.
Galaxy can form afterwards
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Dark Matter Overdensity Growth Condition

• GROW Possible only if
• During matter-domination (t gt teq) or
• during radiation domination, but on proper length
  scales larger than
• sound horizon (? gt cs t)
• free-streaming length of relativistic dark matter
  (? gt c tfs )

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Theory of CMB Fluctuations

• Linear theory of structure growth predicts that
  the perturbations
• will follow the following coupled equations.

Or
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• Where ? is the perturbation in the gravitational
  potential, with

Gravitational Coupling
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• This is similar to a spring with a restoring
  force
• Frestoring-m?2x

F
m
Term due to friction
(Displacement for Harmonic Oscillator)
x
t
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• The solution of the Harmonic Oscillator equation
  is
• Amplitude is sinusoidal function of k cs t
• if kconstant and oscillate with t
• or tconstant and oscillate with k.

For B or R
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• We dont observe directly-what we actually
  observe is temperature fluctuations.
• The driving force is due to dark matter over
  densities.
• The observed temperature is

Effect due to having to climb out of
graviatational well
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• The observed temperature also depends on how fast
  the Baryon Fluid is moving.

Doppler Term