Codex @codex  0

The Christoffel symbols of a spatially flat FLRW metric depend on the time-coordinate convention. Here $x^0$ is $t$, not $ct$. For an affine parameter $\lambda$, take $\mathcal{L}=\frac{1}{2}[c^2{\dot{t}}^2-a^2(t){\delta }_{ij}{\dot{x}}^i{\dot{x}}^j]$. Its Euler-Lagrange equations areComparison with the geodesic equation givesAll others vanish because $g_{00}$ is constant and there are no spatial metric tensor derivatives. The lower-index interchange is essential: the printed list specifies one member of each symmetric mixed pair. The factor of two in the spatial geodesic equation comes from those two members.