Redshift and the scale factor

Builds on: Section 13.1 and Section 4.9
Reading time: 35 min
Exercises: 4
Olympiad problems: 1

The galaxy GN-z11 has a redshift of 10.6, and was once the most distant galaxy known. Its light left it when the Universe was about 430 million years old, and the Lyman α line of hydrogen, which it gave out in the far ultraviolet at 121.6 nm, arrives at 1411 nm, in the infrared. This is why the James Webb Space Telescope, built to find the first galaxies, observes in the infrared. A redshift of 10.6 does not mean that GN-z11 moves at ten times the speed of light. In this section we see what it does mean.

13.2.1The scale factor

Before you readMake a guess first

Light from a galaxy arrives with every wavelength doubled, a redshift of $z = 1$ . What does this tell us?

We describe the expansion by a single number, the scale factor $a (t)$ , which gives the size of the Universe at time $t$ compared with today. Every distance between galaxies far enough apart is proportional to $a$ , and today $a = 1$ . When the Universe was half its present size, $a$ was 0.5.

Light travelling between galaxies is stretched along with the space it crosses. A wave that leaves a galaxy when the scale factor is $a$ arrives today with every wavelength longer by the factor $1 / a$ . The redshift therefore measures how much the Universe has grown since the light set out, and this is the cosmological redshift.

ResultRedshift and the scale factor(13.3)

1 + z = \frac{λ_{obs}}{λ_{emit}} = \frac{1}{a (t_{emit})}

$z$	the redshift of a distant source
$λ_{obs}, λ_{emit}$	the wavelength we observe and the wavelength the source gave out
$a (t_{emit})$	the scale factor when the light left the source, with $a = 1$ today

For small

z

the stretch looks just like a Doppler shift,

z \approx v / c

, which is how Section 13.1 used it. For large

z

it should not be turned into a speed through the Doppler formula, since the light was stretched along the whole of its journey and not at the moment it left.

Figure 13.2Redshift as a stretch

Redshift

Scale factor when the light left
Lyman α and Hα arrive at
Background temperature then

The figure shows a strip of wavelengths from 80 nm to 4000 nm on a logarithmic scale, with the band of visible light hatched. Dotted lines mark the wavelengths of the Lyman α line of hydrogen, 121.6 nm, and of the Hα line, 656.3 nm, as they leave the source, and heavy lines mark where we observe them. On a logarithmic scale every line moves the same distance, since every wavelength is multiplied by the same factor

1 + z

. The table gives the scale factor when the light left and the temperature of the background radiation at that time. The figure can be set to , , where Lyman α arrives in visible light, or .

Quick checkLyman α from a distant galaxy

At what wavelength do we observe the Lyman α line, 121.6 nm at rest, from a galaxy at $z = 3$ ?

13.2.2What else the expansion stretches

Everything that depends on lengths in the Universe changes with the scale factor. The energy of a photon is $h c / λ$ , so it falls as $1 / a$ . The radiation filling the Universe keeps a blackbody spectrum as it is stretched, with every wavelength, including the peak of Wien's law ((4.11)), growing by the same factor, so its temperature falls as $1 / a$ as well. Two wave crests sent a time $Δ t$ apart arrive further apart by the same factor, so every process in a distant galaxy appears slowed. The number of particles in each cubic metre goes as $1 / a^{3}$ , since the same particles once filled a volume $a^{3}$ times as large as today's.

ResultHow quantities change with redshift(13.4)

\begin{aligned} E_{obs} & = \frac{E_{emit}}{1 + z}, T = T_{0} (1 + z) \\ Δ t_{obs} & = (1 + z) Δ t_{emit}, n = n_{0} (1 + z)^{3} \end{aligned}

$E_{obs}, E_{emit}$	the energy of a photon as observed and as given out
$T, T_{0}$	the temperature of the background radiation at redshift $z$ and today, $T_{0} = 2.725$ K
$Δ t_{obs}, Δ t_{emit}$	the length of an event as we see it and as it happened at the source
$n, n_{0}$	the number of particles in each cubic metre at redshift $z$ and today, for particles that are neither made nor destroyed

The slowing of time has been seen directly. Distant supernovae of type Ia rise and fade more slowly than nearby ones, by just the factor

1 + z

Example 13.2How hot the background was

The background radiation now has a temperature of 2.725 K. What was its temperature when the scale factor was 1/1101, at a redshift of 1100?

Solution

By (13.4), $T = 2.725 \times (1 + 1100) = 3000$ K.

AnswerThe radiation was at 3000 K, about the temperature at which hydrogen atoms hold on to their electrons. Section 13.7 shows that this is the moment at which the background radiation we see today set out.

Example 13.3A supernova seen in slow motionfinish it

A type Ia supernova takes 20 days to rise to its peak, as measured by a clock beside it. How long does the rise take as we see it, if the supernova is at $z = 0.5$ ?

The start

Each stage of the rise is stretched in time by the same factor as the wavelengths of its light, which is $1 + z = 1.5$ .

Your stepNow find the observed time of the rise.

days

Common errorsSection 13.2

The cosmological redshift should not be put into the relativistic Doppler formula to find a speed. The light has been stretched all the way along its path, and Section 13.6 shows how to find the speed and distance properly.
The scale factor and the redshift are related by $a = 1 / (1 + z)$ , not by $a = 1 - z$ . At $z = 1$ the Universe was half its present size, and at $z = 3$ a quarter.
The temperature of the background radiation rises into the past, $T = T_{0} (1 + z)$ . Dividing by $1 + z$ in place of multiplying gives a Universe that was colder in the past.
Galaxies and atoms are not stretched by the expansion, so the wavelengths they give out stay the same. Only the light on its way between galaxies is stretched.

ExercisesSection 13.24 questions

13.2.1
What was the scale factor when light now arriving at $z = 1$ set out?
13.2.2
At what wavelength do we observe the Hα line, 656.3 nm at rest, from a galaxy at $z = 0.1$ ?
nm
13.2.3
How many times as many atoms were there in each cubic metre of the Universe at $z = 2$ as today?
times
13.2.4
A photon reaches us from $z = 4$ . What fraction of its original energy does it have?

Olympiad problemsPast IOAA problems, restated1 problem

2011The background radiation in the young Universe10 points

Adapted from IOAA 2011 (Katowice), theory problem 15. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The cosmic background radiation has kept a blackbody spectrum throughout the history of the Universe. Find how its temperature depends on redshift, and give its temperature at $z \approx 10$ , the redshift of the furthest objects then known. The present temperature of the background radiation is 2.73 K.

I solved it without the solution

AnswerThe temperature grows as $T = T_{0} (1 + z)$ , and at $z = 10$ it was about 30.0 K.