Math Exploration 13.2
- Page ID
- 45852
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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:
\[\rho_{crit} = \frac{3H}{8 \pi G} \nonumber \]