Puzzle about proper distance
You can derive the comoving distance formula without the Robertson-Walker metric, but not using &quot;the photon being like a boat on a water stream&quot;. You need to use that the speed of light is constant as peculiar velocity of the photon.
So let $R(t) = a(t) \, r(t)$ be the proper distance. The rate of change is $$\begin{aligned}R'(t) &amp;= a'(t) \, r(t) + a(t) \, r'(t) \\ &amp;= v_{\rm rec} + v_{\rm pec}\end{aligned}$$ where $v_{\rm rec} = a'(r) \, r(t)$ is the recession velocity, and $v_{\rm pec} = a(t) \, r'(t) = \pm c$ is the peculiar velocity. The formula follows.