Split the light of a star or a galaxy by wavelength and dark gaps appear here and there. Atoms along the way have absorbed particular wavelengths, and which wavelengths those are is fixed by the kind of atom. So measuring the same atoms in a laboratory gives a pattern of lines to compare against directly.
Do this with the light of a galaxy and the pattern keeps its shape but sits at longer wavelengths. It is not one line that moves. The whole pattern is stretched by the same proportion.
This shift is written and is called the redshift, with the wavelength measured in the laboratory and the wavelength that arrives from the galaxy. The shift is toward the longer side, so is positive.
While is small it may be read as a speed of recession, . It is the same as an ambulance sounding lower as it drives away: waves from a receding source arrive stretched.
What is needed next is the distance. A speed comes out of one spectrum, but a distance has to be measured by finding an object of known brightness inside the galaxy and comparing it against the brightness that reaches us.
Take the distances and speeds measured this way and set them along the two axes. The points do not scatter. They fall on a single straight line through the origin.
Written out, . The constant is the Hubble constant, and it is the slope of that line. A galaxy twice as far away recedes twice as fast: that is the whole content of the one line.
That the line passes through the origin is what matters. Nearby galaxies recede slowly, and as approaches 0 so does . The speed of recession is fixed by the distance alone, and depends neither on the mass of the galaxy nor on its kind.
If every galaxy is receding, it looks as though we sit at the centre. We do not.
Set galaxies at equal intervals and stretch the ruler between them uniformly. Choose any one of them and watch from there: the neighbour draws away a little, the next one twice as fast, the one beyond it three times as fast. The speed is proportional to the distance.
Choose a different one and nothing changes. From that galaxy too, the farther ones draw away faster. The two rows in the figure differ only in which galaxy was chosen, and the pattern of arrows comes out exactly the same.
The galaxies are not flying through space; the space between them is stretching. So there is no special place, and the same law holds from wherever you look. It is the raisins in a rising dough, every one of them drawing away from every other.
Rewrite as . A distance divided by a speed is a time.
It is the time a galaxy would take to come back to us, running in reverse at the speed it has now. What matters is that this value comes out the same for every galaxy. A distant galaxy has a large distance, but its speed is larger in the same proportion, so the division leaves the same number.
Which is to say that every galaxy set out from one place at one moment. This is called the Hubble time, and it is a rough measure of the age of the universe.
Rough, because the expansion was taken to run at an unchanging rate. Gravity in fact pulls back and dark energy pushes outward, so the rate differs from age to age. Even so, that a single slope yields a time is unchanged.