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If we carry this analogy further then maybe the size and mass of the universe is given by the expression for the Schwarzschild radius of a black hole:

2 * G * M / R = c^2

where M is the mass of the universe and R is the size of the universe.

If you put this expression into the Friedmann equations for the universe, together with the assumption of a flat spacetime, you find:

R = c * t

where R is the size of the universe and t is the age of the universe.

Also one finds a dark energy term such the the density (rho) of dark energy is given by:

rho = (1 / 8 pi G) * 1 / t^2.

This model gives the present density of dark energy rho_0 as

rho_0 = H_0^2 / 8 pi G

Putting in the present Hubble constant H_0 one finds a value of rho_0 which is a few proton masses per m^3.