Temperature at Galactic Scale Perturbation Horizon Entry?

[Kyrios]

Homework Statement

How do I calculate the temperature at which a galactic scale perturbation enters the horizon?
This would be for radiation domination.

Homework Equations

$$\left( \frac{\delta \rho}{\rho} \right)_{\lambda_0} (t) = \left( \frac{a(t)}{a_{eq}} \right) \left( \frac{\delta \rho}{\rho} \right)_{HOR}$$
$$a \propto \frac{1}{T}$$
$$\rho \propto a^{-4} \propto T^4$$

The Attempt at a Solution

length scale of the perturbation is $\lambda_0$ = 1 Mpc
matter-radiation equality perturbation is $\lambda_{0 eq}$ = 100 Mpc
temperature at equality $T_{eq}$ ~ 1 eV

If I do this like for matter domination it gets a little over 8 kev, but I'm not sure how to do it for radiation domination.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?