Cosmological redshift

Light emitted at $t_e$ and observed at $t_0$ has $1+z=a(t_0)/a(t_e)$.

Cosmic expansion stretches an emitted or absorbed spectral feature's wavelength by $1+z$. A small observed wavelength interval at fixed transition wavelength gives a redshift interval by division by the rest wavelength. Cosmological radial distances then require the Hubble parameter and, for proper distance at that epoch, an extra factor $1/(1+z)$.

A local redshift spacing between absorbers corresponds to proper path length $c\Delta z/[H(z)(1+z)]$. Using the same rest-frame transition for all lines gives $\Delta z=\Delta \lambda /{\lambda }_{\mathrm{rest}}$. This conversion assumes cosmological redshift dominates the line spacing and uses a local Hubble-rate approximation.

Redshift drift is the change in a comoving source's cosmological redshift with observer time. It directly probes $\dot{z}=(1+z)H_0-H(z)$.

Differentiating $1+z=a_0/a(t)$ gives

Cosmic chronometers infer the Hubble parameter from differential ages of passively evolving galaxies:They probe $H(z)$ locally in redshift rather than through a distance integral.

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,