 Solution

For a radial null geodesic in the FRW metric, define $\chi (r)={\int }_0^r(1-k{r}^{\prime 2}{)}^{-1/2}d{r}^{\prime }$. Two successive wave crests emitted and received at the same comoving positions obeySubtracting and keeping the first order in the wave periods gives $\Delta t_0/a(t_0)=\Delta t_e/a(t_e)$. Thus the cosmological time dilation and the cosmological redshift have the same factor:Here the emitter and observer are comoving; an additional peculiar velocity would supply a separate kinematic redshift. The derivation uses the null geodesic propagation of the crests rather than interpreting a finite cosmological distance as an ordinary Doppler effect.