Cosmological redshift

Remarks

The definition agrees with the spectroscopic redshift $z = (\lambda_o - \lambda_e)/\lambda_e$, where $\lambda_e$ and $\lambda_o$ are the emitted and observed wavelengths. Indeed, a radial light ray satisfies $d\chi/dt = 1/a(t)$, so two crests that leave the source $\delta t_e$ apart arrive $\delta t_o$ apart with $\delta t_e / a(t_e) = \delta t_o / a(t_o)$. Example: if $a(t_o) = 3\, a(t_e)$, then $z = 2$ and every wavelength has tripled. Non-example: a source moving with respect to the comoving frame has an extra Doppler shift, so its measured redshift is not given by the formula alone. In an expanding model $z > 0$; for nearby sources $z \approx H(t_o)\, D$ to first order, where $D$ is the proper distance and $H = \dot a / a$.