On the derivation of the Kompaneets equation

Abstract

The relaxation of a photon bath to thermal equilibrium via Compton scattering with electrons is described in the Kompaneets equation ( 1956). The equation is mostly known from studies of astrophysical plasmas, for its convergence to the Planck distribution and for explaining the Sunyaev-Zeldovich effect changing the apparent brightness of the cosmic microwave background radiation. We revisit its derivation emphasizing its structure as a Kramers-Moyal diffusion approximation to the quantum Boltzmann equation or Master equation with stimulated emission. We do not assume that the Planck law is stationary in performing the continuum approximation but we emphasize the necessity of the flux or Møller factor to arrive at a continuity equation. On the other hand, the structure allows more general assumptions than originally envisioned by Kompaneets.

Elastic scattering of cosmological gravitational wave backgrounds: primordial black holes and stellar objects

“…Ref. [35] gives us a prescription for assigning a characteristic energy and hence an effective temperature to a population that is out of thermal equilibrium. The closest thing to an effective temperature for PBHs is their Hawking temperature.…”

An SZ-like effect on cosmological gravitational wave backgrounds

“…Conventions for the curvature tensors, covariant and Lie derivatives are all taken from Carroll [27]. We closely follow the work done in [28] while also elucidating on certain details left implicit. All values for cosmological parameters are taken from [29].…”

Simplified derivation of the Kompaneets equation

“…The Kompaneets equation describes the thermal equilibration of radiation via Compton scattering [1][2][3] in a rarefied plasma in which electrons are thermalized at a temperature T . Under assumptions summarized below, this equation for the average number n(ω, t) of photons at frequency ω at time t is ∂n(ω, t) ∂t = n e σ T mc…”

“…Its derivation is "non-trivial since it has to take account of the interchange of energy between the photons and electrons and also include induced effects which become important when the occupation number n is large " [6]. In view of its long history, it is hardly surprising that it has been derived by different methods, typically based on a Fokker-Planck approximation to a Boltzmann or master equation [3]. The term proportional to n 2 (ω) + n(ω) is associated with Compton scattering, including induced scattering, whereas the term proportional to ∂n/∂ω describes a photon diffusion in frequency space and is attributable to "the heating of the photons by hot electrons" [5].…”