Show how the geodesic equations and hence the Christoffel symbols $\Gamma^{a} b c$ can be obtained from a Lagrangian.

In units with $c=1$, the FLRW spacetime line element is

$$d s^{2}=-d t^{2}+a^{2}(t)\left(d x^{2}+d y^{2}+d z^{2}\right)$$

Show that $\Gamma_{01}^{1}=\dot{a} / a$.

You are given that, for the above metric, $G_{0}{ }^{0}=-3 \dot{a}^{2} / a^{2}$ and $G_{1}^{1}=-2 \ddot{a} / a-\dot{a}^{2} / a^{2}$, where $G_{a}^{b}$ is the Einstein tensor, which is diagonal. Verify by direct calculation that $\nabla_{b} G_{a}^{b}=0$.

Solve the vacuum Einstein equations in the presence of a cosmological constant to determine the form of $a(t)$.