KAHIBARO
Table of Contents
Light stretched by the expansion of space
Cosmological redshift is the increase in the wavelength of light caused by the expansion of the universe. When light travels across cosmic distances, the space through which it moves can expand while the light is on its way. As a result, the wavelength becomes longer by the time the light reaches us.
A longer wavelength means the light is shifted toward the red end of the spectrum, because red light has a longer wavelength than blue light. This is why the effect is called redshift.
This idea is different from ordinary motion through space in a simple way. In cosmological redshift, it is not just that a galaxy moves away like a car driving down a road. Instead, the large scale structure of space itself changes with time, and the light is stretched along with it.
Defining redshift
The redshift is represented by the symbol $z$. It compares the observed wavelength of light to the wavelength that was originally emitted.
$$
z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}
$$
If the observed wavelength is larger than the emitted wavelength, then $z > 0$, and the light is redshifted.
This can also be written as
$$
\lambda_{\text{observed}} = (1+z)\lambda_{\text{emitted}}
$$
Since frequency and wavelength are related inversely, the frequency becomes smaller when the wavelength becomes larger.
$$
f_{\text{observed}} = \frac{f_{\text{emitted}}}{1+z}
$$
Important relations for cosmological redshift:
$$
z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}
$$
$$
\lambda_{\text{observed}} = (1+z)\lambda_{\text{emitted}}
$$
$$
f_{\text{observed}} = \frac{f_{\text{emitted}}}{1+z}
$$
A positive redshift means the wavelength has been stretched.
Connection with the scale factor
In cosmology, the size of the universe at a given time is described by the scale factor, usually written as $a(t)$. Cosmological redshift is directly connected to how much the universe expanded while the light was traveling.
If light was emitted when the scale factor was $a_{\text{emit}}$ and observed when the scale factor is $a_{\text{obs}}$, then
$$
1+z = \frac{a_{\text{obs}}}{a_{\text{emit}}}
$$
By convention, the present universe is often assigned
$$
a_{\text{obs}} = 1
$$
so that
$$
1+z = \frac{1}{a_{\text{emit}}}
\qquad \text{and} \qquad
a_{\text{emit}} = \frac{1}{1+z}
$$
This means that large redshift corresponds to light emitted when the universe was smaller than it is today.
For example, if $z = 2$, then
$$
a_{\text{emit}} = \frac{1}{1+2} = \frac{1}{3}
$$
So the universe had a scale factor one third of its present value when that light was emitted.
A simple interpretation
Imagine drawing a wave on a rubber sheet. If the sheet is stretched, the distance between wave crests increases. The wave itself becomes longer. Cosmological redshift works in a similar way. As the universe expands, the wavelength of traveling light stretches.
Spectral lines and measurement
Astronomers measure cosmological redshift by looking at spectral lines. Atoms and molecules emit or absorb light at specific wavelengths. These known wavelengths act like markers. If the same pattern appears at longer wavelengths in the light from a distant galaxy, the amount of shift gives the redshift.
For instance, if a spectral line is known in the laboratory to have wavelength $\lambda_{\text{emitted}}$, and in a galaxy it is observed at $\lambda_{\text{observed}}$, then $z$ follows immediately from the formula.
Suppose a line emitted at $500 \, \text{nm}$ is observed at $650 \, \text{nm}$. Then
$$
z = \frac{650 - 500}{500} = \frac{150}{500} = 0.30
$$
So the redshift is $0.30$.
What redshift tells us
Cosmological redshift tells us that the universe has expanded during the travel time of the light. It also lets astronomers look into the past. Light from very distant galaxies has taken a long time to arrive, so observing high redshift objects means observing the universe at earlier times.
The larger the redshift, the earlier in cosmic history the light was emitted. This makes redshift one of the most important tools in cosmology.
Redshift compared with blueshift
If light is stretched, it is redshifted. If light is compressed, it is blueshifted, meaning the wavelength becomes shorter. In the large scale universe, distant galaxies usually show redshift because the universe is expanding. In some local situations, objects can be blueshifted if they move toward us strongly enough.
A simple comparison is shown below.
| Effect | Wavelength | Frequency | Sign of $z$ |
|---|---|---|---|
| Redshift | Increases | Decreases | $z>0$ |
| No shift | Unchanged | Unchanged | $z=0$ |
| Blueshift | Decreases | Increases | $z<0$ |
For cosmological redshift, bigger $z$ means light was emitted when the universe was smaller.
$$
1+z = \frac{a_{\text{obs}}}{a_{\text{emit}}}
$$
If today $a_{\text{obs}} = 1$, then
$$
a_{\text{emit}} = \frac{1}{1+z}
$$
A note about very small redshifts
For nearby galaxies, small redshifts are often approximately related to recession speed by
$$
z \approx \frac{v}{c}
$$
when $v \ll c$. Here $c$ is the speed of light. This is only an approximation for small redshift. For large cosmological distances, the full expanding universe description is needed.
So cosmological redshift is best understood as a sign of cosmic expansion, not just as a simple everyday Doppler effect.
Summary
Cosmological redshift is the stretching of light due to the expansion of the universe. It is measured with
$$
z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}
$$
and it is related to the cosmic scale factor by
$$
1+z = \frac{a_{\text{obs}}}{a_{\text{emit}}}
$$
A higher redshift means a longer wavelength, a lower frequency, and light emitted when the universe was younger and smaller.