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The Boltzmann EquationsThe Boltzmann Equations
The rate of change in the abundance of a
given particle Unintegrated Boltzmann equation:
Govern the evolution of perturbation in theuniverse Photons
Cold dark matter Baryons
Massless neutrinos
Collision term
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For the Harmonic OscillatorFor the Harmonic Oscillator
One simple example
Nonrelativistic harmonic oscillator Collisionless Boltzmann equation
Equilibrium distribution
f(p,x)=fEQ(E)
C[f]=0
How rapidly the oscillator moves in real space.
How quickly particles lose momentum.Equilibrium distribution
0
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For PhotonsFor Photons
FRW metric of the zero-order homogeneous,
flat universe
Only scalar perturbation here
Vector perturbation:
Overdense region:
Called the conformal Newtonian gauge
Conformal Newtonian Gauge
Newtonian potential
Perturbation to the spatial curvature
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Boltzmann equation for photons
Bose-Einstein distribution
For PhotonsFor PhotonsCollisionless Terms
Hydrodynamics Lo
se energy in expanding universe
The effect of under-/overdense regions
inhomogeneities
The continuity and Euler equationsintegrated
anisotropies
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For PhotonsFor PhotonsCollisionless: Zero/first-Order Equation
Zero-order equation
WithoutNo zero-order collision terms
First-order equationThe wavelength getting stretched as the universe expands
H(t)
Effect of gravityFree streaming
anisotropies
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For PhotonsFor Photons
The scattering process
Collision term
Collision Terms: Compton Scattering
ignore(1) Angular dependence (1% accuracy)(2) Polarization due to Compton scattering (CH10)
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For PhotonsFor Photons
Monopole part of the perturbations
Strong scattering means that the mean free
path of a photon is very small
Collision Terms: Monopole Compton Scattering
0Only monopole perturbation surviveswhen Compton scattering is very efficient.
The temperature on the sky is uniform.
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For PhotonsFor Photons
Boltzmann equation
Conformal time
Fourier transform
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For Cold Dark MatterFor Cold Dark Matter
Collisionless Boltzmann equation for
nonrelativistic matter:
Zero-order term
CH1: an obvious ramification of the expansionCh2: conservation of the energy momentum tensor
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For Cold Dark MatterFor Cold Dark Matter
First-order equation
First-order perturbation
Density VelocityFT
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For BaryonsFor Baryons
Baryons: protons and electrons
Coulomb scattering rate (e+p e+p)> Expansion rate
Overdensities and velocities:
Density
Velocity
FT
QQ, qq symmetriccep antisymmetic
0
Electrons lost
0If conserved
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For BaryonsFor Baryons
Density
Velocity
The evolution of the baryon density
dipole
monopole
Pl is the Legendre polynomial of order l
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SummarySummary
Photon
Cold darkmatter
Baryon
Masslessneutrino
Angular dependence of Compton scattering Temperature field
Sourced by quadrupole
Neutrino distribution
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