Why does the CMB have a spectrum like a black-body radiation?

Question

Equilibrium distributions of particles (Maxwell, Boltzmann, Saha) are achieved by the particle collisions. On the other hand, photons do not interact with each other. From the introductory course in the theory of radiative transfer in astrophysics, I learned that the equilibrium distribution of radiation is obtained through the interactions of photons with atoms (consecutive absorptions and emissions) in an isolated system at a constant temperature (Planck's distribution for black-body radiation). These absorptions and emissions are actually true or thermal absorptions/emissions (inelastic collisions of photons with matter) in which there is a conversion of electromagnetic energy into the internal energy of the gas (plasma) and vice versa (see the book of Mihalas: Stellar Atmospheres). With the help of these processes, a local equilibrium is established between the radiation field and the state of the gas.

If I understood that correctly, then the following is not clear to me. In the book 'Atomic Astrophysics and Spectroscopy' by Pradhan and Nahar (2011, Cambridge University Press) is written:

At the earliest times, radiation and matter were coupled in the sense that photons scatter from free matter particles via Thomson or Compton scattering, and have short mean free paths. Since all radiation energy was thus ‘trapped’, the Universe was in a radiation-dominated state and essentially opaque. The conditions would have been as in an ideal black body characterized by a radiation temperature and a Planck distribution. That would correspond to an extremely hot radiation background, the forerunner of the much cooler present-day cosmic microwave background (CMB).

I don't understand here what are the actual physical processes of the interaction of photons and matter that are responsible for the establishment of the Planck distribution. Thompson/Compton scattering? If so, I don't understand how scattering processes can be responsible for the establishment of equilibrium distribution of radiation. Am I missing something?

Answer 1

Yes, you are missing the fact that at high enough energies, it's possible for a sufficiently energetic photon to create an electron-positron pair out of the vacuum; during the big bang, those particles were being continually created and destroyed in a dense bath of energetic photons- which trended towards thermal equilibrium and hence a blackbody spectrum. That trend was established because the creation/destruction process was fast compared to the expansion rate, and so thermal equilibrium was almost perfect at each successive instant of the expansion.

Answer 2

The CMB we see today was generated when the hydrogen produced from the Big Bang changed from a plasma of protons and electrons to neutral hydrogen atoms. Neutral hydrogen is mostly transparent to radiation with an energy less than 13.6 eV while plasmas interact strongly with light, so the equilibrium between photons and matter was established in the plasma just before recombination occurred.

The interaction of the photons and the plasma is a basically simple process. The electric field of the photons interacts strongly with the electric charge of the free electrons and this allows energy to be transferred between them e.g. by Compton scattering. Hence the energy per degree of freedom of the photons and electrons will quickly become the same.

Answer 3

There are lots of processes that serve to exchange energy between the photons and the particles. Thomson scattering results in no energy change for the light, so that isn't one of them, it just serves to significantly decrease the effective photon mean free path with respect to one of the energy-transferring processes.

The processes that do exchange energy between the photons and the gas particles would include Compton scattering and its inverse process (the upscattering of photon energies by energetic particles), thermal bremsstrahlung and inverse bremsstrahlung, which involve the acceleration of charged particles and the emission/absorption of photons, photorecombination of electrons with protons and helium ions and the photoelectric effect on any atoms/ions with bound electrons. Compton scattering is little different to Thomson scattering in plasmas with temperatures of only 3000K because the photons have energies much less than the rest-mass energy of an electron.

The important point is that each of these processes has an inverse process and in a complete thermodynamic equilibrium, these are in detailed balance and similar arguments to those usually applied to a toy two-level atom, lead to relationships between the emission and absorption coefficients and the requirement that the radiation field is the Planck function at the same temperature as the matter.

Another factor to consider is that there are $10^9$ photons for every baryon in the universe. It is the baryons (and electrons) that are forced to come into equilibrium at the same temperature as the radiation. The blackbody form of the radiation was already established at earlier epochs when the photons were still being created by pair annihilation reactions and then cools as the inverse of the scale factor as the universe expands.