Where Did I Go Wrong in My Big Bang Nucleosynthesis Calculation?

[Niles]

[SOLVED] Astrophysics homework: Cosmology

Homework Statement

I have to show that during the Big Bang Nuchleosynthesis, the scalefactor is approximately given by

$$a(t) = \left( {4H_0^2 \Omega _{r,0} } \right)^{1/4} \sqrt t$$

The Attempt at a Solution

Ok, since the BBNS is during the first ~3 minutes after Big Bang, we are in a flat radiation-dominated Universe. The Friedmann equation takes the form:

$$\frac{{H^2 }}{{H_0^2 }} = \frac{{\Omega _{r,0} }}{{a^4 }}$$

I rewrite and integrate this and I get that

$$a(t) = \left( {3tH_0 \sqrt {\Omega _{r,0} } } \right)^{1/3}$$

Can you guys find my error? This problem seems quite straight-forward, but I can't find my error.

 

[2Tesla]

I think you forgot this:

$$H = \frac{{1}}{{a}} \frac{{da}}{{dt}}$$

or else you lost track of another one of the a's during your integration.

 

[Niles]

That's right, I forgot that. I worked it out now - it really is straight-forward.

Thanks.