Hubble’s discovery. In 1929 Edwin Hubble measured the distances to a few dozen galaxies, and found that the light from nearly all of them is redshifted: the spectral lines are at longer wavelengths than they are in the laboratory. If this is a Doppler shift, the galaxies are moving away from us, and Hubble found that the further away a galaxy is, the faster it moves away:

v = H₀ d

where H₀ is the Hubble constant, about 70 km/s for every megaparsec (Mpc) of distance (1 Mpc = 3.26 million light years). A galaxy 100 Mpc away is moving away at about 7000 km/s, and one 200 Mpc away at 14,000 km/s.

Not an explosion from a centre. This does not mean that we are at the centre of the universe. In the raisin bread model, the dough rises and every raisin moves away from every other raisin. Take any raisin as your home: the nearest raisins move away slowly and the far ones move away fast, in proportion to their distance. (Click “Make the chosen galaxy my home” and see that the picture looks the same.) On the surface of a balloon it is the same: the galaxies are dots on the rubber and the surface stretches. There is no centre of the surface, and no edge. (The centre of the balloon is not part of the universe.) The galaxies do not move through space much: space itself is expanding.

Redshift is a stretching. A light wave travelling through expanding space is stretched along with it, so its wavelength grows in proportion to the scale factor a:

1 + z = λ_observed / λ_emitted = a_now / a_then

For small z, z ≈ v/c. A galaxy that sent out its light when the universe was half its present size has z = 1.

The age of the universe. If the galaxies have always moved at their present speeds, they were all together a time t = d/v = 1/H₀ ago. This is the “Hubble time”:

t = 1 / H₀ = 977.8 / H₀ billion years (with H₀ in km/s per Mpc)

For H₀ = 70 this gives 14.0 billion years, close to the best estimate of 13.8 billion years (the expansion speeded up and slowed down in the past, and the two effects nearly cancel). Hubble’s own value was about 500 km/s per Mpc, which gives only 2 billion years: younger than the Earth. That was a puzzle for years, and the reason is that Hubble’s value was wrong (see the next paragraph).

Why Hubble’s value was wrong. His speeds were good, because redshifts can be measured very accurately. His distances were too small, so for each galaxy the speed divided by the distance came out too big. The distance scale had to be corrected in steps: (1) Two kinds of Cepheid (Baade, 1952). Hubble used Cepheid variable stars as standard candles: the longer the period of a Cepheid, the brighter it is, so its apparent brightness tells you its distance. Baade found that there are two kinds, and the ones in other galaxies are about 4 times brighter than the ones used to calibrate the scale. A star that is 4 times brighter looks as bright as a star twice as far away, so all distances doubled and H₀ halved to about 250. (2) Gas clouds, not stars (Sandage, 1958). Beyond the reach of Cepheids, Hubble used the brightest “stars” in each galaxy as standard candles. Many of these were really huge glowing clouds of gas (HII regions), much brighter than any star, so the galaxies were about 3 times further away again: H₀ fell to about 75. (3) Better and more methods (since then). Cepheids calibrated by parallax, the tip of the red giant branch and Type Ia supernovae give about 70 km/s per Mpc. (4) Too close. Hubble’s galaxies were all near us. Even at 4 Mpc the expansion is only 300 km/s, the same as the random motions of the galaxies themselves (Andromeda is moving towards us), so for the nearest galaxies the pattern is hidden. The idea was right, as Hubble and Milton Humason showed in 1931 with much more distant galaxies, but the number was wrong. In the interactive the correction is shown as the same speeds with the distances multiplied by 2, then by about 6.7 (500 ÷ 75) and by about 7.1 (500 ÷ 70): these are simplified, because the real corrections were made in many steps over several decades.

What the model leaves out. The raisin bread is a flat patch that repeats at its edges, and all the galaxies are the same distance from a flat screen. The data on the graph are made up (drawn from a straight line with random scatter) to show how the method works: they are not Hubble’s own measurements. Speeds are calculated as v = H₀d (the speed of the galaxy as measured in the same way for every home), and the redshift is shown as z = v/c, which is only good for z much less than 1.

Things to try. (1) Choose a galaxy, then double the scale factor. What happens to its distance, speed and redshift? (2) Choose a different home. What is the same, and what is different? (3) Switch to the balloon. Where is the centre? (4) On the graph, find the best-fit line. What age does it give for the universe? (5) Increase the scatter. How well can you still measure H₀? (6) Use the 1929 data: what age do you get, and how does it compare with the age of the Earth (4.54 billion years)? (7) With the 1929 data, press “Why was it wrong?” and step through the corrections. What does the best-fit H₀ become after each one? (8) Use the distance-scale buttons, then “Find the best fit”. If every distance is multiplied by 2 and the speeds stay the same, what happens to H₀?