= Cosmological redshift
{title2=$1+z=a(t_0)/a(t_e)$}
{wiki}
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,
$$
1+z=\frac{\lambda_0}{\lambda_e}=\frac{a(t_0)}{a(t_e)},\qquad
\frac{\nu_0}{\nu_e}=\frac{1}{1+z}.
$$
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>:
$$
\chi=\int_{t_e}^{t_0}\frac{c\,dt}{a(t)}.
$$
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.
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