Redshift drift ($\dot{z}$)

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

The Sandage-Loeb test seeks the spectroscopic velocity drift of stable cosmological absorption features over an observer-time baseline. Many Lyman-alpha forest lines in bright distant quasars can be combined, but wavelength calibration, instrumental stability and peculiar accelerations must be controlled below the tiny cosmological signal. The absorption-template proposal is developed in the original 1998 redshift-drift study.

For a transition with fixed rest wavelength, ${\lambda }_{\mathrm{obs}}=(1+z){\lambda }_{\mathrm{rest}}$. The apparent spectroscopic velocity shift is $dv=cd{\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{obs}}=cdz/(1+z)$. It is a convenient wavelength-shift unit, not necessarily a change in a source's peculiar velocity. Omitting the denominator $1+z$ overestimates the signal at high redshift.

For positive matter and vacuum fractions with ${\Omega }_m+{\Omega }_{\Lambda }=1$, redshift drift vanishes when $s^2={\Omega }_m{s}^3+{\Omega }_{\Lambda }$, $s=1+z$. Factoring out the trivial observer root $s=1$ leaves ${\Omega }_m{s}^2-{\Omega }_{\Lambda }s-{\Omega }_{\Lambda }=0$, giving the displayed positive root of a polynomial. It lies at a positive redshift exactly when ${\Omega }_m<2/3$. It is independent of the Hubble constant at fixed density fractions and is not the redshift of the acceleration-to-deceleration transition.