Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 9B (Cosmology)

Spatial homogeneity says that the velocity of $B$ relative to $A$ depends only on their separation. Comparing both galaxies with $C$ givesThus $\mathbf{v}(\mathbf{x}-\mathbf{y})=\mathbf{v}(\mathbf{x})-\mathbf{v}(\mathbf{y})$, so continuity of the macroscopic velocity field turns this Cauchy functional equation into a linear map:Isotropy requires $RH=HR$ for every rotation matrix $R$. The only such rank-two tensor is a scalar multiple of the identity, and henceThis is the Hubble law. A comoving galaxy has physical position $\mathbf{r}(t)=a(t)\mathbf{x}$, so

The physical particle horizon isFor an Einstein-de Sitter universe, $a(t)\propto t^{2/3}$, soSince $H_0=2/(3t_0)$,The Horizon problem is that widely separated regions of the observed universe have nearly identical conditions even though, in a decelerating standard Big Bang history, their past light cones do not overlap early enough to establish causal thermal equilibrium.