AstroSpeedrun
13.1

The expanding universe

Builds on
Sections 4.3 and 4.9
Reading time
35 min
Exercises
4
Olympiad problems
2

In 1929 Edwin Hubble plotted the speeds of 24 galaxies, measured from the redshifts of their spectra, against their distances, and found that the further a galaxy is, the faster it moves away from us. Georges Lemaître had found the same from fewer data two years earlier and explained it as the expansion of the Universe. Hubble's own value for the rate was about 500 km/s for each megaparsec of distance, seven times today's, because his distances were too small. In this section we see what the law means, why it does not put us at the centre of anything, and what it says about the age of the Universe.

13.1.1Every galaxy sees the same

Before you readMake a guess first

Every distant galaxy moves away from us, faster the further it is. What does this say about our place in the Universe?

When all distances in the Universe grow by the same factor, a galaxy twice as far away moves away twice as fast, since its distance grows by twice as much in the same time. This holds wherever the observer is. The Universe has no centre of expansion and no edge from which galaxies fly outwards. The space between the galaxies grows, much as the spacing of raisins grows in a rising loaf, although the galaxies themselves, held together by their own gravity, do not grow. The figure shows this.

Figure 13.1Every galaxy sees the same expansion
Watch
Distance from O to G
Speed of G away from O
The figure shows a field of galaxies round an observer's galaxy O. Each galaxy has a line ahead of it showing its velocity relative to O, which points straight away from O and is longer for galaxies further away. As the scale factor a grows, every distance grows in proportion, and the drawing keeps O still. Tapping any galaxy makes it the observer, and the same pattern of lines appears round it. The table gives the distance and speed of the marked galaxy G for a Hubble constant of 70 km/s/Mpc today, with the Hubble parameter of that time when the scale factor is not 1. The figure can be set to , , or .

The speed of a galaxy away from us, called its recession speed, grows in proportion to its distance. This is the Hubble–Lemaître law, and the constant of proportion is the Hubble constant H0.

ResultThe Hubble–Lemaître law(13.1)
v=H0d
vthe recession speed of a galaxy
dits present distance
H0the Hubble constant, the present rate of expansion
The law holds for galaxies far enough away that their own motions, a few hundred km/s, are small next to the expansion. Measurements give H0 between about 67 and 73 km/s/Mpc, and the two main methods still disagree. This course uses 70 km/s/Mpc unless a problem gives another value.

For a galaxy close enough that its redshift z, the fractional shift of its lines ((S.30)), is much less than 1, the recession speed is v=cz. The distance then follows from the redshift alone, d=cz/H0. Section 13.2 shows what the redshift means for larger values.

Quick checkA galaxy 100 Mpc away

How fast does a galaxy 100 Mpc away move away from us, if H0=70 km/s/Mpc?

km/s
Example 13.1A distance from a redshift

A galaxy's spectrum shows a redshift of z=0.0040. How fast is it moving away, and how far away is it, if H0=70 km/s/Mpc?

Solution

The redshift is small, so v=cz=2.998×105×0.0040=1199 km/s. By (13.1), d=v/H0=1199/70=17.1 Mpc.

AnswerThe galaxy moves away at 1199 km/s and is about 17.1 Mpc away, near the distance of the Virgo cluster. At such a small distance the galaxy's own motion, which can be a few hundred km/s, makes this uncertain by tens of per cent.

13.1.2The Hubble time and the Hubble distance

If every galaxy had always moved away from us at its present speed, all of them would have been together at the same moment, a time d/v=1/H0 ago. This time is the Hubble time. The distance at which the Hubble law gives the speed of light, c/H0, is the Hubble distance.

ResultThe Hubble time and the Hubble distance(13.2)
tH=1H0dH=cH0
tHthe Hubble time, a rough age of the Universe
dHthe Hubble distance, the distance at which the Hubble law gives the speed of light
cthe speed of light
To find tH in years, H0 must first be put in s⁻¹, with 1 Mpc =3.086×1019 km. For H0=70 km/s/Mpc, H0=2.27×10−18 s⁻¹, tH=13.97 billion years and dH=4283 Mpc. The true age differs from tH because the expansion has not run at a steady rate, which Section 13.5 takes up.

Hubble's own value of 500 km/s/Mpc gave a Hubble time of only 1.96 billion years, less than the age of Earth from its rocks. The conflict lasted until the 1950s, when better distances lowered the Hubble constant.

13.1.3Why the night sky is dark

If the Universe were infinite, unchanging and filled with stars, every line of sight would end on the surface of a star, and the whole night sky would be as bright as the surface of the Sun. This is Olbers' paradox, after Heinrich Olbers, who wrote about it in the 1820s. The sky is dark because the Universe has a finite age. We can see only light that has had time to reach us, and stars have shone for only about 13 billion years, far too short for the average line of sight to reach a star, as the problem below shows. The expansion also redshifts the light of distant galaxies, which dims it further.

Common errorsSection 13.1
  1. The Hubble constant has units of km/s per Mpc, which is an inverse time. To find a time from it, the megaparsec must be turned into kilometres so that the units cancel.
  2. The expansion has no centre. Every observer in every galaxy sees all distant galaxies moving away according to the same law.
  3. Galaxies, stars and planets do not expand. They are held together by their own gravity, and only the distances between widely separated galaxies grow.
  4. The law v=cz holds only for redshifts much less than 1. For larger redshifts it gives speeds faster than light, and the redshift has to be read as a stretch of space, which Section 13.2 explains.
ExercisesSection 13.14 questions
  1. 13.1.1

    A galaxy has a redshift of 0.010. Take H0=70 km/s/Mpc. How far away is it?

    Mpc
  2. 13.1.2

    Two galaxies are seen in opposite directions, each 50 Mpc from us. How fast do they move apart from each other, if H0=70 km/s/Mpc?

  3. 13.1.3

    What is the Hubble time for H0=70 km/s/Mpc? One megaparsec is 3.086×1019 km and one year is 3.156×107 s.

    billion years
  4. 13.1.4

    If the Hubble constant were larger than we think, how would the Hubble time change?

Olympiad problemsPast IOAA problems, restated2 problems
2013The Hubble constant from one galaxy10 points

Adapted from IOAA 2013 (Volos), theory problem 5. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The spectrum of a galaxy whose distance has been measured as 41.67 Mpc shows the Balmer Hα line, which is at 656.3 nm in the laboratory, shifted to 662.9 nm.

(a) Use this to find the Hubble constant. [7 points]

(b) From your result, estimate the Hubble time. [3 points]

Answer(a) H0=72.4 km/s/Mpc. (b) The Hubble time is about 13.5 billion years.

2015How far a line of sight reaches10 points

Adapted from IOAA 2015 (Magelang), theory problem 14. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Suppose we lived in a static, infinitely large universe in which the average number of stars is n=109 per cubic megaparsec and every star has the radius of the Sun, 6.957×108 m. Assume that ordinary Euclidean geometry holds. How far, on average, could we see in any direction before the line of sight met a star? Give the answer in Mpc.

AnswerThe average line of sight would reach 6.26×1017 Mpc before meeting a star. Light takes about 2.04×1024 years to cross that distance, far longer than the age of the Universe, so in our Universe most lines of sight reach no star, which is why the night sky is dark.