A Summing Up — Question 4

Question 4

A small matter excess can survive almost complete pair annihilation

A toy comoving volume initially contains $N_b=10^9+1$ baryons and $N_{\bar b}=10^9$ antibaryons. Each annihilation removes one of each and produces two photons. Ignore all initial photons and entropy redistribution.

1. Find surviving baryon count and produced photon count after complete annihilation of the minority species.

2. Calculate final baryon-to-photon ratio and the initial fractional baryon excess relative to total massive particle count.

Question 4 - Solution

A small matter excess can survive almost complete pair annihilation

(a) Method and calculation

$$\boxed{N_b-N_{\bar b}=1\ \text{is preserved},\quad N_{b,f}=1}.$$$$\boxed{N_{\gamma,f}=2(10^9)=2\times10^9}.$$

(b) Interpretation and further calculation

$$\boxed{\eta_f=N_{b,f}/N_{\gamma,f}=5.0\times10^{-10}}.$$$$\boxed{\frac{N_b-N_{\bar b}}{N_b+N_{\bar b}}=\frac1{2\times10^9+1}\simeq5.0\times10^{-10}}.$$

Physical check

The model demonstrates survival of a pre-existing asymmetry, not its origin. Real annihilation products and photon entropy require a thermal calculation.