One difficulty that we have is that we can’t actually measure the speed of a distant object. We can measure its distance through a variety of proxies, like how bright/faint it is or how large/small it appears on the sky, presuming we know or can figure out how intrinsically bright or large it is. We can also measure its redshift, or how the light is “shifted” from how it would be if we were in the precise location and under the same precise conditions where the light was emitted. That shift, because of our familiarity with how waves shift due to the Doppler effect (such as for sound waves), is something we often translate into a recession speed.
However, we’re not measuring an actual speed; we’re measuring the cumulative effects of motions plus the effect of the expanding Universe. When we say “the Universe is accelerating,” what we actually mean — and this is not what you’d intuit at all — is that if you watch the same object as the Universe expands, it will not only continue to increase in distance from you, getting farther and farther away, but the light that you receive from this object will continue to display an ever-increasing redshift, which makes it seem as though it’s accelerating away from you.
In reality, though, the redshift is due to the expansion of space, not to the galaxy speeding away from you faster and faster. The expansion rate, if we were to actually measure that over time, is still decreasing, and will eventually asymptote to a finite, positive, and non-zero value; that’s what it means to live in a dark energy-dominated Universe.
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