# Miss understanding related with the energy density of radiation in the context of cosmology

The usual definition of radiation energy density in the context of statistical physics is given by $$U=a_{B}T^{4}$$

With $$a_{B}=7.5657\times 10^{-16} J m^{-3} K^{-4}$$. So $$U$$ has units of $$J m^{-3}$$

On the other hand I read in some General relativity textbooks that the parameter $$\rho$$ (the parameter that appears in the Friedmann equations) is the energy density, but if I look at the units in the of the Friedmann equation for the Hubble parameter

$$H^{2}=\frac{8 \pi G }{3}\rho$$

I find that $$\rho$$ has units of $$kg/m^3$$. So in the particular case of radiation $$\rho_{r}$$ don't have the same units as $$U$$, then $$\rho_{r}$$ is not the energy density of radiation?

In astrophysics and cosmology it is common to omit factors of $$c$$, where $$c$$ is the speed of light. This means that in these units an energy density will look like a mass density. To fix this, you have to put in a factor $$c^2$$.
• I know that the correct relation is $U=c^{2}\rho_{r}$, but I was looking for some reference. However your answer makes more clear my problem, thank you. Commented May 28, 2020 at 5:51