 Cosmic microwave background diffusion damping

Cosmic microwave background diffusion damping, or Silk damping, damps the photon-baryon acoustic oscillator on small scales because Thomson scattering gives photons a finite mean free path. Repeated scatterings mix hotter and colder regions. Departures from perfect tight coupling also appear as photon-baryon velocity slip and a photon quadrupole, producing heat-conduction and shear-viscosity damping.

In natural units, the positive conformal scattering rate is ${\Gamma }_T=a{n}_e{\sigma }_T$, with proper free-electron number density $n_e$. A random-walk estimate gives squared comoving diffusion distance of order $\int d\eta /{\Gamma }_T$. More precisely, the leading polarization-inclusive tight-coupling amplitude containswhere $R=3{\rho }_b/(4{\rho }_{\gamma })$ is the baryon loading parameter. The corresponding power spectrum suppression is $e^{-2k^2/k_D^2}$. The approximation applies while the expansion in scattering time is controlled; it cannot be extended through a transparent universe by simply integrating $1/{\Gamma }_T$ indefinitely.^[ref]

Diffusion accumulates before last scattering and becomes especially important as the free-electron population falls. The finite width of the cosmological visibility function can also average oscillating line-of-sight sources; that averaging is a distinct mechanism acting in the same epoch. Both effects help shape the small-angular-scale damping tail, whose scale probes the ionization and baryon history.