Free-streaming photon Boltzmann equation

In the absence of collisions, a photon phase-space density is constant along its geodesic. Linearizing this statement gives a first-order transport equation for the direction-dependent temperature perturbation.

The line-of-sight solution integrates the gravitational source along the unperturbed photon trajectory. In Fourier space the integrating factor $e^{ik\mu \eta }$ accounts for free streaming between emission and observation.

At linear order in Newtonian gauge, free streaming, gravitational redshift, and Thomson scattering combine into a first-order transport equation for the photon temperature perturbation. The collision term drives the radiation toward its monopole plus the electron bulk-velocity dipole.

The optical depth from conformal time $\eta$ to observation at ${\eta }_0$ isThe factor $e^{-\tau (\eta )}$ is the probability that a photon travels from $\eta$ to the observer without another scattering.

The visibility functionis the probability density for the conformal time of a CMB photon's last scattering.

The CMB line-of-sight solution separates sharply localized last-scattering sources, weighted by the cosmological visibility function, from integrated gravitational sources weighted by $e^{-\tau }$ along the photon trajectory.