Deriving the reheating equation from the first law of thermodynamics

[pleasehelpmeno]

Homework Statement

i i am trying to derive $\dot{\rho_{R}}+4H\rho_{R}-\Gamma_{\phi}\rho_{\phi}=0$ as in Kolb an turner (Boltzmann describing reheating).

Is the correct approach to, use the 1st law dU=dQ-pdV, but what would dQ be?

 

[clamtrox]

That is just the equation of conservation of energy, with an extra term with transfers energy from the inflaton to radiation.

 

[pleasehelpmeno]

how though can it be derived?
I no that: $dU=(a^{3}\rho_{R})$ and $1/3\rho dV= pdV$, but I am not sure about dQ.

It is obviously then $\rho_R d(a^3)+a^{3}d\rho_{R}=dQ - 1/3\rho_{R}d(a^3)$ Then by dividing through by dt one can get $\dot{\rho}_{R}+4H\rho_{R} -\frac{dQ}{dt} =0$.

I don't see how one can justify $\Gamma_{\phi}\rho_{\phi}=\frac{dQ}{dt}$

 

[clamtrox]

I don't see how you can justify using the first law in that form, when clearly in reheating the number of particles is not conserved. It's my understanding that the $\Gamma \rho$-term is just phenomenological.