In the 1920s, working the great 100-inch telescope on Mount Wilson, Edwin Hubble was doing something no one had managed before: measuring the distances to the faint, fuzzy "spiral nebulae" and pinning down that they were whole galaxies far outside our own Milky Way. Then he paired each distance with the galaxy's redshift — how far the dark lines in its spectrum had slid toward the red end — and a startling pattern jumped out of the data.

Almost every galaxy is rushing away from us. And not randomly: the farther a galaxy lies, the faster it recedes, in clean proportion. Double the distance, double the speed. There is only one honest reading of that fact — the universe itself is expanding, carrying the galaxies apart like markings on a stretching sheet of rubber. This page builds that idea from the observation up: Hubble's law, the scale factor that measures the stretch, the cosmological redshift it produces, and the Friedmann equations that govern how the whole thing evolves.

Hubble's law: farther means faster

Hubble's discovery compresses into one line. If a galaxy sits a distance d away and recedes with velocity v, then

where H_0 — the Hubble constant — is the constant of proportionality, the same for every galaxy. It is usually quoted in the slightly odd-looking units of \text{km/s per megaparsec}:

meaning a galaxy one megaparsec away (1\ \text{Mpc} \approx 3.26 million light-years) recedes at about 70\ \text{km/s}; one at two megaparsecs at 140\ \text{km/s}; and so on. The "constant" is constant across space today, not across time — its present-day value is what the little 0 denotes.

That last point is the one worth pausing on, and the diagram below makes it concrete.

No centre: the raisin bread and the balloon

If everything is flying away from us, surely we must be at the centre of the explosion? No — and this is the deep and beautiful part. Picture a lump of raisin-bread dough rising in the oven. As it swells, every pair of raisins moves apart. Stand on any one raisin and look around: nearby raisins drift away slowly, distant raisins drift away fast (there is more expanding dough between you and them), and the recession speed is proportional to separation — Hubble's law, exactly. Yet no raisin is special. Each one sees itself as "the centre" and all the others receding. There is no centre.

The two-dimensional version is the surface of an inflating balloon with dots inked on it. Blow it up and every dot recedes from every other dot; the fabric between them stretches. The dots aren't crawling across the rubber — the rubber itself is growing. And the balloon's surface has no edge and no centre-point on the surface: the "centre" of a balloon is off the surface entirely, in a direction that doesn't exist for a flat creature living in the rubber. That is the right mental picture for our universe: space itself is expanding, galaxies ride along, and asking "where is the centre?" is like asking which point on the balloon's skin is the middle.

The scale factor: putting a number on the stretch

To make the stretching-rubber picture quantitative, cosmologists factor every physical distance into two pieces. Each galaxy is given a fixed comoving coordinate \chi — a label that rides along with the expansion and never changes, like the fixed grid-lines painted on the balloon. All the growth is bundled into a single function of time, the scale factor a(t). The actual, physical distance to that galaxy is

By convention we set a(t_0) = 1 today, so a = 0.5 was an epoch when everything was half as far apart, and a = 2 will be a future when distances have doubled. Expansion is simply a(t) growing. Differentiate the distance and watch Hubble's law fall straight out:

Compare with v = H_0 d and you can read off what the Hubble "constant" really is — the fractional growth rate of space, evaluated now:

So Hubble's law isn't an approximate empirical fit that happens to be linear; for the smooth, large-scale expansion it is exact, a mathematical identity of the scale-factor picture. And the crucial reinterpretation: the galaxies are not hurtling through space away from a blast site. They sit at fixed comoving coordinates while the space between them stretches. The recession is the stretching of space itself.

Cosmological redshift: light stretched by the expansion

Now we can explain the redshift Hubble measured — and explain why it is not the everyday Doppler shift of a receding siren. A light wave is a pattern of crests and troughs riding in space. While that wave is in flight across billions of years, space itself expands, and the wave is carried along and stretched with it, exactly like the dots on the balloon. Its wavelength grows in lockstep with the scale factor. If the light is emitted when the scale factor is a(t_\text{emit}) and received when it is a(t_\text{obs}), then

The redshift z is defined as the fractional stretch, z = (\lambda_\text{obs} - \lambda_\text{emit})/\lambda_\text{emit}, which rearranges to the clean, central result:

Reveal the figure to watch a single wave leave a galaxy, ride the growing space, and reach us longer than it started.

The Friedmann equations: what governs the stretch

Hubble's law tells us the universe is expanding now; it doesn't say how the expansion rate changes with time. That comes from feeding the assumption of a smooth, uniform universe into Einstein's general relativity. The result is the two Friedmann equations, the master equations of cosmology. The first governs the expansion rate:

You do not need to derive it to read what each term is doing to H = \dot a/a:

• Density \rho — all the matter and radiation. Its gravity pulls inward, working to decelerate the expansion. In an otherwise empty, matter-filled universe, expansion would coast and slow.
• Curvature k — the geometry of space: k = +1 a closed (spherical) universe that may recollapse, k = 0 a flat one, k = -1 an open (saddle-like) one that expands forever. It enters as a 1/a^2 term that dilutes as space grows.
• Cosmological constant \Lambda — the energy of empty space, dark energy. Unlike matter it does not dilute as space grows, and it pushes outward, accelerating the expansion. It dominates the late universe.

The second (acceleration) Friedmann equation makes the tug-of-war explicit, and reveals a twist — pressure gravitates too:

The sign of \ddot a decides whether the expansion is speeding up or slowing down. Ordinary matter and radiation (with \rho + 3p/c^2 > 0) make \ddot a < 0 — gravity brakes the expansion. The \Lambda term pushes the other way. Remarkably, dark energy behaves as though it has a large negative pressure, so it lands on the accelerating side. Since the late 1990s we have known the expansion is in fact accelerating — the universe is \Lambda-dominated today.

Critical density: is the universe flat?

Set \Lambda = 0 for a moment and ask: how much density would make space exactly flat (k = 0)? Putting k = 0 in the first Friedmann equation and solving for \rho gives the critical density:

a ludicrously thin gruel — about five hydrogen atoms per cubic metre. We compare any real density to it with the density parameter

where \Omega > 1 means a closed universe, \Omega < 1 open, and \Omega = 1 exactly flat. Counting all the contributions — ordinary matter, dark matter, radiation, and dark energy — observations find the total \Omega_\text{total} \approx 1 to within a percent. As far as we can measure, the universe is flat, and its energy budget is dominated by dark energy (\Lambda), which is why it is now accelerating.

Hubble time and the age of the universe

The Hubble constant secretly carries a timescale and a length scale. Since H_0 has units of 1/\text{time}, its reciprocal is a time — the Hubble time — and multiplying by c gives a length, the Hubble distance:

The Hubble time is a rough estimate of the age of the universe (rough because H was different in the past). Let us actually evaluate it. First strip the megaparsec out of H_0. Using 1\ \text{Mpc} = 3.086 \times 10^{19}\ \text{km}:

That this crude "one over the expansion rate" lands right next to the measured age of the universe (13.8 billion years) is no accident — it is one of the first clues that the expansion picture is telling the truth. And the Hubble distance d_H = c/H_0 \approx 4{,}300\ \text{Mpc} \approx 14 billion light-years marks roughly the scale beyond which recession outruns light.

Worked examples

Example 1 — recession from distance. A galaxy sits d = 100\ \text{Mpc} away. With H_0 = 70\ \text{km/s/Mpc}, Hubble's law gives its recession speed directly:

About 2.3\% of the speed of light — comfortably in the Doppler-like regime, where z \approx v/c is a good approximation.

Example 2 — redshift from the scale factor. We observe a galaxy whose light left when the universe was half its present size, so a_\text{emit} = 0.5. Its redshift follows from 1 + z = a_\text{obs}/a_\text{emit} with a_\text{obs} = 1:

Every spectral line arrives at twice its emitted wavelength: a line emitted at \lambda_\text{emit} = 500\ \text{nm} (green) reaches us at \lambda_\text{obs} = (1+z)\lambda_\text{emit} = 1000\ \text{nm} — shifted clean out of the visible into the infrared.

Example 3 — distance implied by a velocity. A quasar's spectrum shows a recession velocity of v = 21{,}000\ \text{km/s}. Turning Hubble's law around, d = v/H_0, gives a rough distance:

This is exactly how redshift becomes a cosmic measuring tape: measure a galaxy's redshift, convert to velocity, divide by H_0, and read off how far away it is.