Modren Cosmology

8/3/2019 Modren Cosmology

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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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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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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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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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