In that case, how are galaxies meant to collide? Surely if they were all constantly speeding up that should never happen? And if gravity is meant to be such a weak force, how could it slow down a galaxy?
This is where I find the rubber sheet analogy better than the balloon. Imagine the universe as a large, smooth (meaning completely frictionless) rubber sheet. A mass in the universe is just like a mass placed on the sheet. It creates a dent in the sheet, but is still free to move around on it if given a push. Another object placed on the sheet and pushed near the first one will follow a curved path due to the dent, and may end up orbiting it or even hitting it depending on the exact conditions. This is the basic analogy for explaining gravity in relativity.
Now, to make it relevant to this discussion, imagine the sheet is no longer just sitting there, but is constantly being stretched. This is completely independent of any objects that may be resting on it, it's the sheet itself that is stretching. If you now take two objects placed a long way from each other so that they don't notice their respective dents, they will each observe the other moving away. This is the case even though neither is moving with respect to the sheet.
Now take many such objects, all far enough from each other that their dents don't affect each other. Every single object will observe every other object moving away from it. Since the stretching is perfectly even, the speed the other objects seem to be moving is proportional to the distance between them. To every object, it appears that it is stationary and every thing else is moving away from it, which must logically mean that it is at the centre. However, if you now consider the rubber sheet to be curved into a sphere, rather like the balloon analogy, you can see that there is no centre - every part of the rubber sheet is equivalent to every other part.
If you now move the objects on the sheet closer together, so that they start being influenced by the dents caused by other objects on the sheet, you can see the effects of gravity. The sheet is still expanding, but if the dent is big enough and the objects close enough together, they will fall towards each other faster than the sheet pulls them apart. Depending on the details, this may result in a collision, or it may just lead to them orbiting each other. If you have more than two objects near each other, the motion can get quite complex - even with just three objects we don't currently know how to solve the equations of motion except in a few specific cases.
Obviously, if you do have a group of objects that all fall towards each other, that means they can't be falling towards any objects somewhere else on the sheet. So what you end up with is clusters of objects held together by their mutual attraction, but with all the clusters moving away from each other due to the continued stretching of the sheet. Of course, since the dent caused by the combined group of objects is much larger than that from a single object, the clusters themselves can interacy with each other and form even larger scale structures - studying this is one of the focuses of modern cosmology.
All this assumes that the rubber sheet is being stretched at a constant rate. That means that once a couple of objects are orbiting each other, they're pretty much going to stay that way unless another one comes along and disrupts things. However, if someone starts stretching the sheet faster and faster, it will reach a point where the stretching of the rubber between the two objects is actually faster than the speed they are falling towards each other (orbit is bascially just falling that keeps missing), and the objects will be pulled apart. Since the speed the stretching makes objects move apart depends on the distance between them, the large scale clusters will be the first things to come apart. As the stretching accelerates further, smaller clusters will come apart until eventually every object has been pulled so far from any others that it can no longer even see they exist.
So on that point you are almost right. If the stretching is fast enough, gravity can't pull galaxies together to collide. However, the important point is that accelerating expansion doesn't necessarily mean fast expansion. It will
eventually be much faster if it continues to accelerate, but it hasn't reached the point where it can pull galaxies away from their neighbours yet.
So, does that help at all?
Please explain how the differences in time at the big bang have an impact on the current universe. If it does not have an effect, then isn't it just mental masterbation? Why not start talking about how many angles can dance on the head of a pin or other famous questions?
Fundamental physics can actually be quite important. The current universe is a direct result of whatever happened before, so understanding what happened before leads to better understanding of what is happening now. And what is happening now is important because, aside from knowledge for its own sake, it leads to things like nuclear fission, hopefully fusion soon and who knows what else after that.
Anyway, everyone knows that angles don't dance. They're not
acute enough.
