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

Question

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?

Answer 1

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 $c^2$.