BBN: How is the expansion rate related to the freeze-out of Neutrons?

I've been making my way through several papers and textbooks on the subject, but they all seem to gloss over this subject: how is the equilibrium of protons, neutrons and photons related to the expansion rate?

From what I've picked up on the subject, photons, protons and neutrons are all floating around in a hot soup. Protons are relatively stable, but neutrons decay in about 15 minutes. While the soup is dense, this isn't a problem as the photons readily interact with protons making and destroying neutrons in equal amounts.

But after the universe has expanded by a certain amount, the cross-section of the proton becomes diminished with respect to the size of the universe and the probability of a photon interacting with it drops. When the probability drops beyond a certain amount, new neutrons stop being created and we only have 15 minutes or so for the protons to join with the neutrons and become nuclei (either Deuterium, Helium or Lithium), or they decay into protons.

I'm sure I've got some of this wrong, so could someone please explain in plain English how the expansion rate of the Universe determines the ratio of Hydrogen to Helium. And what would happen if the expansion rate were slower or faster than FLRW.

• The faster the expansion, the sooner the freeze-out happens and it can be shown to happen at a higher temperature (temperature does drop faster but as you go back in time the temperature would be higher and the latter effect dominates). Then at a higher temperature the equilibrium ratio between neutrons and protons is given by the Boltzmann factor $\exp(-\Delta m c^2/(k T))$ where $\Delta m$ is the mass ratio between neutrons and protons. The higher the temperature, the closer the ratio will be to 50-50. – Count Iblis Dec 17 '18 at 2:54
• I've seen several mentions of a cross-section which has units of $cm^3 s^{-1}$. Intuitively I think of the cross section of a proton and the probability of a random photon hitting the proton and creating a neutron. How does the this factor into the conversion ratio and the freeze-out temperature? – Quarkly Dec 17 '18 at 17:13