In a solution set I was reading for cosmology I saw that the energy density of radiation is defined as $\epsilon_r=\epsilon_c-\epsilon_p$. Where $\epsilon_p$ is the energy density of particles at that time and $\epsilon_c$ is the critical density of that time. Is this something we can always say is true and if so why ?
We can write $\epsilon_c=\epsilon_m+\epsilon_r$ as $$\epsilon_c/\epsilon_c=\epsilon_m/\epsilon_c+\epsilon_r/\epsilon_c$$
In this universe, we should assume that $\kappa=0$ and $\Lambda=0$. So we are left with a flat universe and total density is equal to the critical density. So these are the conditions that we can say this is true. In other words, this is true for a hypothetical universe, We don't know the values for $\Omega_m$ or $\Omega_r$.
Is this something we can always say is true and if so why?
Under some conditions, which I described above, we can say its true. A simple example is that for our universe we cannot say such a thing.