Acceleration means something crossing space. Light speed through space is a constant, it never accelerates beyond that. Matter can absorb and slow down photons, but they max out at their top speed in a vacuum.
The ongoing expansion of space is still an unsolved mystery in science. We have not established a cause. At one point in the early history of the Universe, there was a rapid inflation that was faster than light could cross space, putting some distant galaxies permanently outside of the range of our observation; as a result we will never know how much of existence is beyond our field of view because their light can never reach us. So it's common to refer to known space that we can see as the Observable Universe.
"I'm planning to live forever. So far, that's working perfectly." Steven Wright