18.1.1: Math Exploration 13.2

When we calculate escape velocity, we set the total energy equal to zero. That is equivalent to setting the curvature term in the Friedmann equation to zero:

$$\frac{k c^2}{S^2} = 0 \nonumber$$

The Friedmann equation then becomes:

$$H^2 - \frac{8 \pi G \rho}{3} = 0 \nonumber$$

The only two adjustable quantities in the equation now are $ρ$, the average density of the Universe, and the expansion rate, $H$. Solving for $ρ$ in terms of $H$ we get:

$$\rho_{crit} = \frac{3H}{8 \pi G} \nonumber$$