A type Ia supernova puts out very nearly a fixed amount of light. Measure how bright it looks and the distance follows. It is the top rung of the distance ladder.
Plot these supernovae by redshift against apparent faintness. The nearby ones fall along the line drawn from Hubble's law, as they should.
Measured out past of about , the points left the line. They are fainter than the brightness predicted for their redshift, which is to say they are further away than expected.
The offset is small, some to magnitudes. But it grows with distance, and it always runs the same way. Measurement error does not line up like that.
How the expansion has run shows up in a single curve, . Here is the size of space, taken as now.
If the contents were matter alone, gravity only ever pulls, so the expansion would have to slow. The curve would bend downward.
That distant supernovae are fainter than predicted means the path to a given redshift is longer. That happens when the expansion along the way ran faster than predicted.
The curve that fits bends upward. Far from slowing, the expansion is speeding up. Something is pushing space apart against gravity, and it goes by the name of dark energy.
It has not been speeding up all along, though. Back when the density of matter was high, gravity had the upper hand.
Expansion thins matter out. The same contents spread through a larger space, so the density falls in inverse proportion to the cube of the size of space.
Dark energy does not thin out. It is a property of space itself, so however far space expands, its density stays exactly what it was.
Sooner or later, then, the two must trade places. And they did: at , some billion years ago, the expansion turned from slowing to speeding up. That is where the curve changes the way it bends.
Set the densities side by side against the size of space. Matter goes as , falling away to the right.
Dark energy, , runs flat. Let space grow to twice the size and there is twice as much of it, because the density never changes.
A falling line and a flat line must cross. The universe passes from an age ruled by matter to an age ruled by dark energy, and once it has passed there is no going back.
The turn to acceleration comes slightly before that crossing. What speeds the expansion up is not the densities being equal, but dark energy exceeding half the density of matter. The dashed line in the figure marks that moment.