(published on 05/24/11)
Figuring out the acceleration requires a careful comparison of the redshifts and the apparent distances, based on brightness. Say that there was no acceleration. Then for smallish distances, the redshift and the distance would be exactly proportional. For larger distances, there's a slightly more complicated relativistic formula, but still just a calculation of how distance should depend on redshift. So to judge acceleration people look at the differences from that calculated curve. For smallish redshifts, objects seem to be too close, as if they weren't moving away that quickly until recently. That means the expansion has been accelerating. For big redshifts, the effect reverses, indicating that the expansion was decelerating a few billion years ago. That's just what's expected if there's a constant background acceleration. Recently, with the matter whose gravity causes deceleration being all spread out, the acceleration wins. When the matter was more concentrated, the deceleration was winning.
Here's a figure showing how the aparent distance depends on redshift, and then showing the small deviation from what would be expected in a no-acceleration picture.
(published on 08/17/13)