Red shift tells us that something is moving away from us. This is because that, as something moves away from us, it “stretches” the light out, shifting it toward the lower (red) end of the spectrum.
So how do we distinguish between something that is just moving away from us as a result of its own momentum vs something moving away from us because the universe is expanding?
When we look at many different things in the night sky, the degree to which they are moving from us is correlated with distance; things that are further away are moving away faster. This is highly indicative of universal expansion, as if it was simply objects moving under their own power, we wouldn’t expect there to be any correlation between them and distance.
Not to mention that everything* is moving away from everything else. If it was just random motion, we would expect some things moving away and some things moving toward us.
Lastly, without expansion, any movement away from us would gradually slow as a result of gravity. Yet not only is the motion not slowing down, it is actually speeding up! Why? Well, we don’t know that yet.
* – Not strictly “everything” is moving away from everything else. Within the scale of a few million light years, gravity is still strong enough to keep things gravitationally together and counteract the expansion of space. But outside that, everything is moving away from everything else.
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