Time is what we call the ticking of a clock, but the ticking of the clock is merely motion. The fundamental character of the universe is motion. It came into existence with the BB because that's what the BB was, motion. I would guess that if the universe were to cease expanding, it would cease to exist. Another way of putting it is that all motion within the universe, right down to the sub-atomic level, is derived, made possible, from the expansion of the universe. It seems logical that if all motion ceased to exist, all matter and energy would cease to exist as well, since motion is fundamental to the internal dynamics of all matter and energy. But, in fact, the universe can't stop expanding.
We are told now that the rate of expansion is greater at large distances than at nearer distances. I wouldn't call this an acceleration of expansion, but an acceleration of collapse. It could be that the universe blew through an enormous black hole in another universe upon it's creation, and is now heading toward it's own enormous black hole of annihilation. We see this as an expansive movement, but probably because of a distortion in our perception of space/time. This idea would need to be studied by someone with a much greater mathematical talent than I have.