bogie said:
I think it safe to say the if accelerating expansion is the case, that galaxies would appear to be moving away from each other at an accelerating rate.
This is a reasonable statement, with some caveats about the 'moving away' part, since thinking entirely kinematically about receding galaxies implies that they can move apart at greater than the speed of light, which can't be explained kinematically. This question however is a can-o-worms that has been opened in many other threads, so it's probably best not to confuse this present discussion with it! So yes, in basic terms your statement that the galaxies appear to be moving away from each other at an increasing rate is reasonable.
My question is, if accelerating expansion is the case, can we conclude that in terms of closed, flat or open, the universe would be open? In other words, there is no indication in theory that deceleration would ever occur given the case of acceleration?
The simple closed (universe collapses in big crunch), open (expansion rate goes to a linear relationship with t) and flat (the balance line between the two) picture you may have seen is only valid for matter only universes. In this case the universe model depends only on the total amount of mass relativity to a critical value.
You cannot get acceleration in this picture at all, since matter will not accelerate the expansion, you need dark energy to do this (in the standard picture anyway) Once you add dark energy into the picture the universe models become more complex, since it depends not only on the amount of dark energy but also its properties. So you could have a flat universe that has a big crunch or a big rip or a closed universe that never re-collapses etc etc. It all depends on the properties of the dark energy.
This attached image may help to guide your understanding of our current best guess model of the universe. What it shows is that initially the universe was in fact deccelerating as matter was the dominant energy component. At a certain time (roughly 5-6 Billion years ago I think, somewhere around there) dark energy became the dominant component in the universe, and began to cause the expansion to accelerate. So the theory does in fact have a deccelerating and accelerating phase. Once you add dark energy the flat, closed and open models are not a very useful tool to understanding the dynamics of the expansion.
The reason that matter dominates early on and dark energy later is that the density of dark energy stays roughly the same for all time (if dark energy is a cosmological constant it stays exactly the same for all time). By contrast the density of matter drops with the inverse cube of the scale factor of the universe, since if you double the volume of the universe, but still only have the same amount of matter then the density will halve. Therefore the greater the total expansion since t=0 the more dominant dark energy will be over matter. Whichever component has the greater density at a given time will dominate the dynamics of the universe, early on matter, later dark energy.
Another way to think about this might be that if dark energy is vacuum energy, i.e. some inherent energy associated with space itself (which is a possible candidate for dark energy, and almost equivalent to a cosmological constant) then as the universe expands there is more space in the universe hence more dark energy in total. By contrast the total amount of matter stays the same, reducing the matter density as the expansion proceeds.