Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 10D / a / Solution

A comoving coordinate $\mathbf{\chi }$ labels a galaxy carried by the homogeneous cosmological flow. Its physical separation at time $t$ is $\mathbf{r}(t)=a(t)\mathbf{\chi }$, where $a$ is the scale factor. Thus $\mathbf{v}=\dot{a}\mathbf{\chi }=(\dot{a}/a)\mathbf{r}$, the Hubble law, apart from peculiar velocities relative to this flow. In particular