When using the Friedmann equation (flat space, no cosmological constant): H = sqrt (8 pi G / 3 ) * rho, if we use rho in mass/volume, H is in (time)^-1 like it should. Now for some inflation models, we use: H = sqrt (8 pi G / 3 ) * V(Phi). It seems that V(Phi) should also be able to be converted to mass/volume. In chaotic inflation models, the function V(Phi) = ½ m^2 Phi^2 is often used. I know “natural units” are employed in these theories, but I was wondering if there is a way to convert V(Phi) = ½ m^2 Phi^2 into units of mass/volume?
Inability to ascribe accurate values for mass/volume is the usual reason this calculation is not attempted very often. We can derive a rough estimate for mass density in the universe, but is highly model dependent. Working backwards from such estimates has been the usual approach, AFAIK.