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I'm trying to figure out how to arrive at the final expression, as given in the lecture notes. I tried to work this out by myself but getting different figure and units. I've considered the followings,

normalize the equation from its values e.g. spins and x parameter, as g~100, x~10(m/t~10) also I considered the (G)^1/2 (from m_planck)= 10^19 GeV when solving. Also converted, Critical density = 1.054 h^2 10^-5 GeV/cm^3 (cubed).