The Big Bang — Question 6

Question 6

The deuterium bottleneck depends on the abundance of energetic photons

Use the rough criterion $\eta^{-1}e^{-B_D/(k_BT)}=1$, where baryon-to-photon ratio $\eta=6.0\times10^{-10}$ and deuterium binding energy $B_D=2.225$ MeV. Ignore the polynomial factor in the thermal tail.

1. Solve for the thermal energy at which this criterion is met.

2. Compare with the binding energy and explain why temperature much below BD is needed.

Question 6 - Solution

The deuterium bottleneck depends on the abundance of energetic photons

(a) Method and calculation

$$e^{-B_D/(k_BT)}=\eta\Rightarrow B_D/(k_BT)=\ln(1/\eta).$$$$\boxed{k_BT=2.225/\ln[1/(6.0\times10^{-10})]=.104784\ {\mathrm{MeV}}}.$$

(b) Interpretation and further calculation

$$\boxed{B_D/(k_BT)=21.2341}.$$$$n_{\gamma,>B_D}/n_b\simeq\eta^{-1}e^{-B_D/(k_BT)}.$$

Physical check

The enormous photon-to-baryon ratio keeps a destructive high-energy tail even when the mean thermal scale is far below the binding energy. This is a scale estimate, not a reaction network.