Consider a sphere centered on any galaxy in the universe with arbitrary radius R (where R is big enough that the average density, r, of matter in that sphere is the same as the average density anywhere in the universe). The total mass of that sphere is the volume of the sphere times the density:

M = (4pr/3)R³ . (1)

Now consider a galaxy having some mass m that lies on the surface of the sphere (i.e., at a distance R from the galaxy at the center). The galaxy is attracted to the center of the sphere by all the mass within the sphere. I.e., its potential energy is given by:

E_B = -GMm/R. (2)

E_K = (1/2)mV², (3)

where V, the velocity of the galaxy relative to the center of the sphere, is given by Hubble's Law:

V = H₀R. (4)

E_TOT = E_B + E_K = -G(4pr/3)R³ m/R + (1/2)m(H₀R)². (5)

The closure density is that density for which E_TOT = 0. Setting the right hand side of equation (5) equal to zero, one finds that m and R factor out, giving the result

r_cl = 3H₀²/(8pG). (6)

This is the average density of the universe for which the Hubble expansion gives any pair of galaxies exactly escape velocity with respect to each other, so we call it the critical density, r_cr. This result has the remarkable property is that it applies to any portion of the universe, no matter how large (because R cancels out).

If the actual density, r, is less than r_cr, the mass within the sphere is not enough to stop the expansion (E_TOT > 0), so the galaxies will have escape velocity and the universe will expand forever. On the other hand, if r > r_cr (E_TOT < 0), the galaxies will all fall back on each other and the universe will collapse again.

We define W₀ = r/r_cr. If W₀ > 1, the universe will collapse again. Otherwise, it will expand forever.