Diffusion Damping - Magnitude

Magnitude

The magnitude of diffusion damping is calculated as a damping factor or suppression factor, represented by the symbol, which figures into the Boltzmann equation, an equation which describes the amplitude of perturbations in the CMB. The strength of the diffusion damping is chiefly governed by the distance photons travel before being scattered (diffusion length). What affect the diffusion length are primarily the properties of the plasma in question: different sorts of plasma may experience different sorts of diffusion damping. The evolution of a plasma may also affect the damping process.

Where:

• is the conformal time.
• is the "differential optical depth for Thomson scattering". Thomson scattering is the scattering of electromagnetic radiation (light) by charged particles such as electrons.
• is the wave number of the wave being suppressed.
• is the visibility function.

The damping factor, when factored into the Boltzmann equation for the cosmic microwave background radiation (CMB), reduces the amplitude of perturbations:

Where:

• is the conformal time at decoupling.
• is the "monopole of the photon distribution function"
• is a "gravitational-potential in the Newtonian gauge". The Newtonian gauge is a quantity with importance in the General Theory of Relativity.
• is the effective temperature.

Mathematical calculations of the damping factor depend on, or the effective diffusion scale, which in turn depends on a crucial value, the diffusion length, . The diffusion length relates how far photons travel during diffusion, and comprises a finite number of short steps in random directions. The average of these steps is the Compton mean free path, and is denoted by . As the direction of these steps are randomly taken, is approximately equal to, where is the number of steps the photon takes before the conformal time at decoupling .

The diffusion length increases at recombination because the mean free path does, with less photon scattering occurring; this increases the amount of diffusion and damping. The mean free path increases because the electron ionisation fraction, decreases as ionised hydrogen and helium bind with the free, charged electrons. As this occurs, the mean free path increases proportionally: . That is, the mean free path of the photons is inversely proportional to the electron ionisation fraction and the baryon number density . That means that the more baryons there were, and the more they were ionised, the shorter the average photon could travel before encountering one and being scattered. Small changes to these values before or during recombination can augment the damping effect considerably. This dependence on the baryon density by photon diffusion allows scientists to use analysis of the latter to investigate the former, in addition to the history of ionisation.

The effect of diffusion damping is greatly augmented by the finite width of the surface of last scattering (SLS). The finite width of the SLS means the CMB photons we see were not all emitted at the same time, and the fluctuations we see are not all in phase. It also means that during recombination, the diffusion length changed dramatically, as the ionisation fraction shifted.