Why is the deuterium bottleneck temperature 0.1 MeV?

Question

During big bang nucleosynthesis (BBN), deuterium has a lower binding energy per nucleon (~1.1 MeV) than the other similar nuclei, and so prevents heavy elements from forming until the temperature drops below about 0.1 MeV. Above 0.1 MeV, deuterium is unstable and will be broken apart by photodisintegration. This is the so-called "deuterium bottleneck." My question is, what is the argument for the deuterium bottleneck temperature being a tenth of the binding energy per nucleon?

Answer 1

There are at least two points to explain why photodisintegration dominates until temperatures fall to $k_BT \sim 0.06$ MeV, which is well below the binding energy of the deuteron ($2.2$ MeV).

1. Photon energies in the early universe have a Planck distribution. The Planck distribution peaks at energies of $2.8 k_B T$ and has an approximately exponential "Wien tail" that extends to higher energies as $\sim 2h\nu^3\exp(-h\nu/k_BT)/c^2$, where $h\nu$ is the photon energy.

2. If $k_B T \sim 0.06$ MeV, the peak of the Planck function is at $0.17$ MeV but the number density of photons at $2.2$ MeV would still be suppressed by something like $\sim 10^{-10}$ compared with its peak value. However, the crucial point is that the photon to baryon ratio in the early universe is $\sim 10^9$. Thus even though the fraction of photons with $E>2.2$ MeV is tiny at temperatures of $0.06$ MeV, in terms of absolute numbers they are comparable with the number of baryons.

Answer 2

The Planck distribution of photon energies is

$$ B_\nu \propto \frac{(h\nu)^3}{e^{h\nu/kT}-1} $$

when the photon gas is close enough to thermal equilibrium (at temperature $T$) for the statistics to work. Here $\nu$ is the frequency, so $h\nu$ is the photon energy.

While the most numerous photons (per unit energy) have energy $h\nu \approx 3kT$, the high-energy tail of the distribution means that there are some number of photons with enough energy to photodissociate deuterium even at lower temperatures. The "bottleneck" occurs when the photodissociation rate is faster than the formation rate.