Einstein–de Sitter universe

Example

The Einstein–de Sitter universe is the spatially flat Friedmann–Lemaître–Robertson–Walker metric filled with pressureless dust and with no cosmological constant. In units with $c = 1$, its scale factor, Hubble parameter and density are $$a(t) = \left( \frac{t}{t_0} \right)^{2/3}, \qquad H(t) = \frac{2}{3t}, \qquad \rho(t) = \frac{1}{6\pi G t^2}, \qquad t > 0.$$

Check with the Friedmann equations for $k = 0$, $p = 0$, $\Lambda = 0$. The first reads $H^2 = \tfrac{8\pi G}{3}\rho$: the left side is $\tfrac{4}{9t^2}$ and the right side is $\tfrac{8\pi G}{3} \cdot \tfrac{1}{6\pi G t^2} = \tfrac{4}{9t^2}$. The second reads $\ddot a / a = -\tfrac{4\pi G}{3}\rho$: the left side is $-\tfrac{2}{9t^2}$ and the right side is $-\tfrac{4\pi G}{3} \cdot \tfrac{1}{6\pi G t^2} = -\tfrac{2}{9t^2}$.