Friedmann equations

Remarks

The proof is a computation of the Einstein tensor of $g$. In the comoving frame its only nonzero components are $G^t{}_t = -3(\dot a^2 + k)/a^2$ and $G^i{}_j = -\left( 2\ddot a/a + (\dot a^2 + k)/a^2 \right) \delta^i{}_j$. The time-time component gives the expansion equation. The space-space components, combined with the expansion equation, give the acceleration equation. The continuity equation is the conservation law $\nabla_a T^{ab} = 0$, which follows from the contracted Bianchi identity $\nabla_a G^{ab} = 0$. Where $\dot a \neq 0$ it can also be obtained by differentiating the expansion equation in $t$ and using the acceleration equation. Example: with $k = 0$, $\Lambda = 0$ and dust ($p = 0$), the continuity equation gives $\rho a^3 = \text{const}$, and the expansion equation then gives $a \propto t^{2/3}$. In terms of $H = \dot a / a$, the expansion equation reads $\Omega - 1 = k/(a^2 H^2)$ with $\Omega = 8\pi G \rho/(3H^2) + \Lambda/(3H^2)$, so the sign of $k$ is fixed by whether $\Omega$ exceeds $1$. The equations are named after Alexander Friedmann, who derived them in 1922 for $k > 0$ and in 1924 for $k < 0$.

Going deeper

For a student

Under the cosmological principle, the ten Einstein equations reduce to these two ordinary differential equations for $a(t)$. Together with an equation of state $p = p(\rho)$ they form a closed system. All the standard models, such as the Einstein–de Sitter, de Sitter, Milne and Lambda-CDM models, are solutions of them. The acceleration equation shows that if $\rho + 3p > 0$ and $\Lambda \le 0$, then $\ddot a < 0$, which forces a Big Bang in the finite past of an expanding model.