Friedmann Equation

This derivation is largely based on notes by David Weinberg.

Let us consider a self gravitating sphere of uniform density $\rho$. The radius of the sphere $R$ changes according to Newton's law of gravity

$\frac{d^2 R}{d t^2} \approx - \frac{G \rho R^3}{R^2}$

Multiplying by $d R/dt$ and integrating with respect to time yields

$\dot{R}^2 \approx - G \rho R^2 + C$

Where $C$ is an integration constant. If $R_0$ is some initial radius, then we can define a dimensionless scale factor $a = R/R_0$, and express our equation in terms of this dimensionless constant:

$\frac{\dot{a}^2}{a^2} = -G \rho + \frac{C}{R_0^2a^2}$

The equation we obtained is very similar to the Friedmann equation.