 Solution

Use a radial null geodesic of the FRW metric in the form $d{s}^2=-d{t}^2+a(t{)}^{2}d{\chi }^2$ along a fixed spatial direction, with $c=1$. For an emitter and observer at fixed comoving coordinates, the comoving separation isApply this to two adjacent wave crests. Since their endpoint separation is the same, subtraction to first order gives $\delta t_o/a(t_o)=\delta t_e/a(t_e)$. These intervals are the local periods measured by the comoving observers, so the received wavelength is stretched by $a(t_o)/a(t_e)$. Consequently the cosmological redshift obeyswhen $a(t_o)=1$. This relation assumes comoving emitter and observer; additional peculiar velocities supply kinematic redshift factors.