Cosmological redshift ($1+z=a(t_0)/a(t_e)$)

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.

A redshift estimated from multi-band fluxes or an inferred spectral energy distribution rather than directly resolved spectral-line wavelengths. calibration, templates, dust and population differences can produce biases and catastrophic outliers. In lensing or clustering analyses, the inferred source redshift distribution affects the geometric kernel as well as assignment to redshift bins.

Let the scale factor grow by $f$ from an initially thermal photon distribution. geodesic redshift gives ${\nu }_i=f\nu$, and conserving photons in corresponding comoving frequency bins gives $n_{\nu }(t)=f^{-2}{n}_{f\nu }(t_i)$. Inserting Planck's law leaves the same spectral form at temperature $T_i/f$. Total photon-number conservation alone would not prove that the entire radiation spectrum stays Planckian; frequency redshift and collisionless propagation are also required. Absorption, injection or spectral mixing can invalidate this one-temperature argument.

For a comoving emitter and receiver, photon energy falls as $1/(1+z)$ and arrival intervals grow by $1+z$. Their combination reduces bolometric flux by $(1+z{)}^2$ in addition to the area dilution.

A cosmological redshift relates a particular emission event to a particular reception event along a null geodesic. Two sources' redshifts on Earth's past light cone do not specify the light exchanged between those sources. If a photon from B passes through A before reaching Earth, and A's comparison emission lies at that passage event, then $1+z_{BA}=(1+z_B)/(1+z_A)$. Without this causal and temporal specification the ratio need not give the redshift measured in A.

Frequency ratios multiply when a photon is transformed first between a source and its local comoving frame, then propagated through an expanding universe. Thus the kinematic redshift and cosmological redshift factors multiply. Their redshifts themselves add only to first order when both are small.

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 $\dot{z}=(1+z)H_0-H(z)$.

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.

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,