When Does the Radiation Epoch Begin with a Scale Factor of ~t1/2?

[PeteSampras]

When begin the radiation epoch with scale factor ~t^1/2 ?

I read in some text that this begin at Planck time, but in other texs says that begin approx at 3 minutes.

 

[Buzz Bloom]

Hi Pete:
I am not sure how to define the beginning and end of the radiation epoch, but I will take a crack at it.

The beginning
The radiation epoch must come after the inflation epoch ends, since the scale factor during inflation is

a ≈ e^t.​

Here is a quote from Wikipedia:

The inflationary epoch lasted from 10⁻³⁶ seconds after the conjectured Big Bang singularity to sometime between 10⁻³³ and 10⁻³² seconds after the singularity.
https://en.wikipedia.org/wiki/Inflation_(cosmology) .​

You are correct that the radiation epoch is characterized by

a ≈ t^1/2.​

The end
I don't think there is a well defined criterion for this. The end will occur before ρ_r, the mass-energy density of radiation (photons) equals the mass-energy density of matter ρ_m. The problem is deciding on the value of the ratio

R = ρ_r : ρ_m > 1​

to use for the end of the radiation epoch and the start of the transition to the matter epoch. When that transition ends, the matter epoch is characterized by

a ≈ t^2/3.​

If you specify the value you want to use for R, it is easy to calculate the value of a which corresponds to this R value. Calculating the corresponding value for t is more difficult.

Regards,
Buzz