Cosmological redshift is one of the central observational ideas in modern cosmology. When light travels through an
expanding Universe, its wavelength is stretched along with space. That means an emitted wavelength
\(\lambda_{\mathrm{emit}}\) is later observed as a longer wavelength \(\lambda_{\mathrm{obs}}\) if the source is receding
with the cosmic expansion. The amount of stretching is described by the redshift \(z\).
Redshift from wavelengths
The standard wavelength definition of redshift is
Redshift formula.
\[
\begin{aligned}
z &= \frac{\lambda_{\mathrm{obs}}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}
\end{aligned}
\]
This can also be written as
\[
\begin{aligned}
z &= \frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}}-1
\end{aligned}
\]
so the ratio \(\lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}\) is especially important. If the observed wavelength is
larger than the emitted wavelength, then \(z>0\), which is the usual cosmic redshift case. If the observed wavelength is
smaller, then \(z<0\), corresponding to a blueshift rather than an expansion redshift.
Scale factor
In cosmology, the scale factor \(a\) describes the relative size of the Universe compared with today. By convention,
\(a=1\) today. The relation between redshift and scale factor is
Scale factor relation.
\[
\begin{aligned}
a &= \frac{1}{1+z}
\end{aligned}
\]
Combining this with the wavelength ratio gives another very useful form:
\[
\begin{aligned}
a &= \frac{\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{obs}}}
\end{aligned}
\]
So if light is observed at one and a half times its emitted wavelength, then the scale factor at emission was
\(a = 1/1.5 = 2/3\). In other words, the Universe was about two-thirds of its present size when that light was emitted.
Worked sample
Consider an emission line with
\(\lambda_{\mathrm{emit}} = 400\ \mathrm{nm}\) and
\(\lambda_{\mathrm{obs}} = 600\ \mathrm{nm}\).
The wavelength ratio is
Step 1. Wavelength ratio.
\[
\begin{aligned}
\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}}
&= \frac{600}{400} \\
&= 1.5
\end{aligned}
\]
Step 2. Redshift.
\[
\begin{aligned}
z &= 1.5 - 1 \\
&= 0.5
\end{aligned}
\]
Step 3. Scale factor.
\[
\begin{aligned}
a &= \frac{1}{1+0.5} \\
&= \frac{1}{1.5} \\
&\approx 0.667
\end{aligned}
\]
This means the light was emitted when the Universe was about \(66.7\%\) of its present size in this simplified reading.
Look-back time estimate
A full look-back time requires integrating the cosmological expansion history, which depends on the contents of the Universe.
For a lightweight educational preview, this calculator uses a simple estimate based on the Hubble time:
Hubble time estimate.
\[
\begin{aligned}
h &= \frac{H_0}{100},\\
t_H &\approx \frac{9.778}{h}\ \mathrm{Gyr}
\end{aligned}
\]
Then the look-back time is approximated as
Simple look-back estimate.
\[
\begin{aligned}
t_{L,\mathrm{approx}} &\approx t_H\frac{z}{1+z}
\end{aligned}
\]
With \(H_0 \approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\), the Hubble time is about \(14\ \mathrm{Gyr}\).
For the sample \(z=0.5\), this gives a look-back time of roughly \(4.7\ \mathrm{Gyr}\). This is only a quick estimate,
but it gives the right qualitative idea that larger redshift generally means looking farther back in time.
Physical interpretation
The key point is that cosmological redshift is not just a Doppler shift in the everyday sense. In cosmology, it is tied to
the expansion of space itself. As the Universe expands, wavelengths stretch by the same scale factor, which is why the
relation \(1+z = 1/a\) is so central. The right-hand graph in the calculator shows that as redshift increases, the scale
factor decreases. Very large redshift corresponds to light emitted when the Universe was much smaller than it is today.
Important limitation
This tool is intentionally an educational preview. It does not compute exact cosmological distances or exact look-back times
from a full \(\Lambda\)CDM model, and it does not distinguish carefully between spectroscopic redshift, peculiar-velocity
effects, and detailed relativistic cosmological models. At university level, one would study the Friedmann equations,
comoving distance, luminosity distance, angular-diameter distance, and the exact integral formula for look-back time.
Still, the formulas used here capture the most important first ideas: wavelength stretching gives redshift, and redshift
immediately determines the cosmological scale factor.
| Concept |
Main relation |
Meaning |
| Redshift |
\(z = (\lambda_{\mathrm{obs}}-\lambda_{\mathrm{emit}})/\lambda_{\mathrm{emit}}\) |
Measures how much the wavelength has changed |
| Wavelength ratio |
\(1+z = \lambda_{\mathrm{obs}}/\lambda_{\mathrm{emit}}\) |
Direct stretch factor of the wavelength |
| Scale factor |
\(a = 1/(1+z)\) |
Relative size of the Universe at emission |
| Equivalent form |
\(a = \lambda_{\mathrm{emit}}/\lambda_{\mathrm{obs}}\) |
Scale factor from the wavelength ratio directly |
| Hubble time |
\(t_H \approx 9.778/h\ \mathrm{Gyr}\) |
Rough inverse expansion timescale |
| Look-back estimate |
\(t_{L,\mathrm{approx}} \approx t_H z/(1+z)\) |
Simple educational estimate of how far back in time the light was emitted |