How do we know that galaxies are accelerating away from us and not moving at constant speed? People often point to the observation that the further away a galaxy is, the faster it appears to be moving away from us, implying acceleration.
However, wouldn't we expect to see the same observation even without any acceleration? Imagine there are some objects in space all moving in random directions and speeds, relative to Earth. After long enough time, all objects will appear to be moving away from Earth, even if they were moving towards it initially. And after long enough time, the objects that move fastest should be farthest away, by the simple definition of speed!
In short, even if galaxies weren't accelerating, we would still see that the further away a galaxy is, the faster it recedes.