# Weak interaction cross-section

1. Feb 14, 2007

### Orbital

1. The problem statement, all variables and given/known data
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.

2. Feb 14, 2007

### Dick

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

3. Feb 15, 2007

### Orbital

I cannot find any such formula...

4. Feb 15, 2007

### Dick

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

5. Feb 15, 2007

### Orbital

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

6. Feb 15, 2007

### Dick

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

7. Feb 15, 2007

### Orbital

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?

8. Feb 15, 2007

### Dick

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?

9. Feb 15, 2007

### Orbital

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

10. Feb 15, 2007

### Dick

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

11. Feb 15, 2007

### Orbital

They are essentially the same?

12. Feb 15, 2007

### Dick

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).

13. Feb 15, 2007

### Dick

You know, I think number density is proportional to T^3 regardless of domination.

14. Feb 15, 2007

### Orbital

Uhm, has it got something to do with 3d space?

15. Feb 15, 2007

### Dick

It also has to do with red-shift. E is proportional to 1/a, where a is the scale factor.

Last edited: Feb 15, 2007