A Summing Up — Question 1

Question 1

A relic annihilation rate connects a particle cross section to cosmic dilution

After a species becomes nonrelativistic, assume its conserved number density scales as $n\propto a^{-3}$ and its constant annihilation coefficient is $\langle\sigma v\rangle$. During radiation domination $H\propto a^{-2}$. Initially $\Gamma/H=8$, with $\Gamma=n\langle\sigma v\rangle$.

1. Derive how Gamma/H scales with a and find the expansion factor at which it reaches one.

2. At that point find the number-density and Hubble-rate ratios relative to their initial values.

Question 1 - Solution

A relic annihilation rate connects a particle cross section to cosmic dilution

(a) Method and calculation

$$\Gamma/H\propto a^{-3}/a^{-2}=\boxed{a^{-1}}.$$$$1=8(a_i/a_f)\Rightarrow\boxed{a_f/a_i=8}.$$

(b) Interpretation and further calculation

$$\boxed{n_f/n_i=8^{-3}=1/512}.$$$$\boxed{H_f/H_i=8^{-2}=1/64,\quad(\Gamma/H)_f=8(64/512)=1}.$$

Physical check

Conserved abundance is a post-equilibrium scaling assumption. A full relic calculation evolves annihilation and production together rather than imposing fixed comoving number through freeze-out.