# Weak interaction cross-section

## Homework Statement

Cross-sections for weak interactions at an energy E increase with E as $E^2$. Show that the rate of weak interactions in the early universe depends on the temperature T as $\sigma_{wk} \propto T^5$

2. The attempt at a solution
The only formula I can find is
$$\sigma_{wk} = g_{wk}^2 \left[ \frac{k_B T}{(\bar{h} c)^2} \right]^2$$
where $g_{wk} \approx 1.4 \times 10^{49}$ erg cm^3 is the weak interaction coupling constant.
I have no clue on how to proceed.

Dick
Homework Helper
The 'rate' depends upon more than the cross section. Find a formula for rate.

I cannot find any such formula...

Dick
Homework Helper
Ok. Then we'll have to make one up. What other factors besides cross section would affect the interaction rate?

The number density and the speed of the particles i guess.

Dick
Homework Helper
Hey, you're pretty good at this! Would you say proportional to both and the cross section (at least roughly)?

I'm not sure... I found a formula

$$\tau_{coll} = \frac{1}{n \sigma_{wk} c}$$

where

$$n = 0.2 \left( \frac{k_BT}{\bar{h} c} \right)^3$$

is the number density. This would give

$$\tau_{coll} \propto \frac{1}{T^5}$$

but this is not it, is it?

Dick
Homework Helper
That is it! tau is the inverse of the rate, right? Under what conditions can you use that formula and are they compatible with the description 'early universe'? What happened to our velocity dependence?

Maybe the particles have the speed of light? But what about the energy in the problem statement?

Dick
Homework Helper
Yes, it assumes the particles are relativistic. I'll throw the other question back to you. What's the relation between T and E?

They are essentially the same?

Dick
Homework Helper
Up to a constant (Boltzmann's to be specific), sure. If you understand why number density is proportional to T^3 then I think you have the whole thing (hint: it also assumes the universe is radiation dominated).

Dick