The comoving congruence is geodesic; its expansion is three Hubble parameters per unit conversion constant

(The comoving curves are geodesics.) In the product chart the vector field ${\mathrm{\partial }}_t$ has components ${\delta }^a{}_0$, so for ${\mathrm{\nabla }}_{{\mathrm{\partial }}_t}{\mathrm{\partial }}_t$ the chart expression of [L2] is ${\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}{\delta }^b{}_0{\delta }^c{}_0={\mathrm{\Gamma }}^a{}_{00}=0$ for $a=0,1,2,3$ by [L1]. Hence ${\mathrm{\nabla }}_{{\mathrm{\partial }}_t}{\mathrm{\partial }}_t=0$. Multiplying by $c^{-2}$ and using the function-linearity and Leibniz rules of [L2] gives ${\mathrm{\nabla }}_{n}n=c^{-2}{\mathrm{\nabla }}_{{\mathrm{\partial }}_t}{\mathrm{\partial }}_t=0$. Therefore each comoving curve ${\gamma }_x$ of [L3], whose tangent is ${\mathrm{\partial }}_t=cn$, satisfies ${\mathrm{\nabla }}_{{\gamma }_x^{\mathrm{\prime }}}{\gamma }_x^{\mathrm{\prime }}=c{\mathrm{\nabla }}_n(cn)=c^2{\mathrm{\nabla }}_{n}n=0$: it is a geodesic with affine parameter $t$, and by [L3] its parameter is proper time. If $\mathrm{\Sigma }\mathrm{\ne }\mathrm{\varnothing }$, the nonempty interval $I$ gives a nonempty product $M=I\times \mathrm{\Sigma }$, open in itself, and [L3] supplies the smooth future unit timelike field $n$; thus it is an observer congruence on $M$ in the exact sense of Observer congruences, their observer curves and adapted charts, with vanishing geodesic acceleration as just proved. If $\mathrm{\Sigma }=\mathrm{\varnothing }$, all pointwise tensor and geodesic assertions are vacuous, and no observer-congruence designation is made. This is claim (a).

(Expansion.) In the product chart, $n=c^{-1}{\mathrm{\partial }}_t$ has components $n^0=c^{-1}$ and $n^i=0$, all of them constant as functions on the chart domain, so ${\mathrm{\partial }}_a{n}^a={\mathrm{\partial }}_0{c}^{-1}+{\sum }_i{\mathrm{\partial }}_{i}0=0$. The divergence formula of [L2] therefore gives ${\mathrm{\nabla }}_a{n}^a={\sum }_{a,b}{\mathrm{\Gamma }}^a{}_{ab}{n}^b={\sum }_a{\mathrm{\Gamma }}^a{}_{a0}{c}^{-1}=c^{-1}({\mathrm{\Gamma }}^0{}_{00}+{\mathrm{\Gamma }}^1{}_{10}+{\mathrm{\Gamma }}^2{}_{20}+{\mathrm{\Gamma }}^3{}_{30})$. By [L1], ${\mathrm{\Gamma }}^0{}_{00}=0$ and ${\mathrm{\Gamma }}^i{}_{i0}=H{\delta }^i{}_i=H$ for each $i=1,2,3$ (no sum over $i$ in the symbol, and the three indices each contribute $H$), so ${\mathrm{\nabla }}_a{n}^a=c^{-1}\cdot 3H=3H\mathrm{/}c$. This is claim (b).

(Physical velocity and vanishing acceleration.) Since $U=cn$, linearity gives ${\mathrm{\nabla }}_a{U}^a=c{\mathrm{\nabla }}_a{n}^a=3H$, hence $H=\frac{1}{3}{\mathrm{\nabla }}_a{U}^a$. For the acceleration, $a^b=n^a{\mathrm{\nabla }}_a{n}^b=c^{-2}({\mathrm{\nabla }}_{{\mathrm{\partial }}_t}{\mathrm{\partial }}_t{)}^b=0$ by step 1.1. This is claim (c).

(Conclusion.) Step 1.1 proves the geodesic property and the affine/proper-time reading of claim (a); step 1.2 computes the expansion $\vartheta =3H\mathrm{/}c$ of claim (b); and step 2.1 proves the physical-velocity divergence identity and the vanishing acceleration of claim (c). No step used a field equation or a matter model beyond the definitions recalled in [L3], and the inherited assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is the one carried by the tangent-bundle and connection suppliers.