Why is the baryon to photon ratio assumed to be constant?

For standard BBN calculations, we use the baryon to photon ratio $$\eta=\frac{n_b}{n_\gamma}$$ that we get from CMB. Now, this clearly assumes that $$\eta_{BBN}=\eta_{CMB}.$$ There are around 400000 years in between. Why can we assume that $$\eta$$ stays the same until then? I thought $$\eta$$ goes inversely with the scale factor $$a$$.

The baryon to photon ratio is essentially fixed by the annihilation of particles and anti-particles in the early universe. If there had been absolute symmetry between the numbers of particles and anti-particles, then the ratio would have been zero! Once all the particle/anti-particle annihilation processes have concluded (once photon energies fell below the rest mass of an election-positron pair), then photons outnumbered baryons and electrons by a factor of $$\sim 10^9$$.
The (ionised) matter and radiation at that point are in equilibrium. Thus for every process (e.g. thermal bremsstrahlung) producing a photon, there is a similar rate of the inverse process occurring that removes a photon. This effectively conserves the photon number in a comoving volume. Obviously as expansion takes place, the density of photons decreases, but the baryon density also decreases by exactly the same factor and thus their ratio, $$\eta$$, is constant.
• My question concerns standard BBN calculations. If the ratio $\eta$ is not assumed to stay constant, why can we use the CMB measurement of $\eta$ for BBN calculations (see Schramm-plot for the lithium problem)? My initial thought was also that $\eta$ must have been different at the time of BBN and CMB decoupling. Jun 9 at 11:36