How Did the Author Derive Equation (10) in the Context of Radiation Density?

[shinobi20]

From the paper, https://arxiv.org/abs/astro-ph/0305015, on page 3,
How did the author arrived with equation (10)? By using the radiation density and (8) defined in the previous paragraph,

## ρ_r = \frac{3}{4}Ts ~## (radiation density) I think the author got it wrong (## ρ_r = \frac{4}{3}Ts ~##)
## T\dot s + 3HTs = Γ \dot φ ^2~## (8)

We have, ##~T \dot s = \frac{4}{3} \dot ρ_r ~## and ##~ \frac{4}{3} \dot ρ_r + 4Hρ_r = Γ \dot φ ^2~##

But why is that in the paper it is written as ##~ \dot ρ_r + 4Hρ_r = Γ \dot φ ^2~##?

Also, how did the author get the radiation density ## ρ_r = \frac{3}{4}Ts ~## by saying that we set ##~δm_T = 0~## in (2)?

 

[AndyW]

Thank you for bringing this discrepancy to our attention. After reviewing the paper, we have found that the author has indeed made a mistake in equation (10). The correct equation should be ##~\dot ρ_r + 4Hρ_r = Γ \dot φ ^2~##, as you have pointed out. This error may have been a simple typo or a miscalculation on the author's part.

To answer your second question, the author obtained the radiation density ##ρ_r = \frac{3}{4}Ts## by setting ##δm_T = 0## in equation (2). This is because in the early universe, the matter energy density is dominated by radiation and thus, setting the perturbation in matter energy density to zero effectively eliminates the matter contribution, leaving only the radiation contribution.

We apologize for any confusion this error may have caused and we will make sure to bring it to the attention of the author. Thank you for your careful reading and critical thinking.