# How is the energy density of radiation relatedd to the energy density of particles and the critical density?

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 $$\epsilon_c/\epsilon_c=\epsilon_m/\epsilon_c+\epsilon_r/\epsilon_c$$

$$1=\Omega_m+\Omega_r$$ or

$$\Omega_r=1-\Omega_m$$

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.

• Thanks so much for all your help , I just finished my exam earlier today and I aced it :) – bhapi Jan 11 at 16:18
• Thats great, happy to help – Reign Jan 11 at 16:41