Equating critical energy density to matter density

Question

I'm asked first to calculate the critical energy density $\rho_{crit}$ of the universe given certain constants. The value I got was $5500\,\mathrm{MeV}\,\mathrm{m}^{-3}$.

Then I'm asked to calculate various number densities of particles assuming the universe is made up of a single type. To do this, I'm told I need to set the universe's matter density equal to $\rho_{crit}$.

I'm not sure how to do this since they are in different units. My expression to find $\rho_{crit}$ had a $c^2$ in it, and if I take that out I would have units of matter density. Is that what I'm supposed to do?

Answer 1

By the mass-energy equivalence:

$E = mc^2$

If $E = 5500$ $\mathrm{MeV}$, then:

$m = \frac{E}{c^2} = 9.8 \times 10^{-27}$ $\mathrm{kg}$.