if we know the rate at which the " universe " expanded after the big bang?

Yes, it possible to calculate how big (or old) the universe was when the forces of " Dark Energy " become evident.

(there is not enough matter in the universe to stop the galaxies flying away)

There is evidence (from distant supernovas) of Dark Energy having an influence from about 5 billion years after the big bang.

But it could reasonably be assumed that dark-energy was present from the moments right after the big-bang.

And it may be that dark-energy is somehow linked to cosmic inflation (the two may be the same force, or not).

"In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density."

http://en.wikipedia.org/wiki/Inflation_%28cosmolog...

Look up "Inflation Theory" for information on these ideas. It is commonly still referred to as The Big Bang Theory as part of that chain of ideas under that same name, but it is a new field with new ideas related to what the model is and will be. It begins with all matter (or pre-matter rather) crunched in a ball and then expanding faster than the speed of light, etc.. It is a fictional way of imagining how things might have come to be as they are today, but it is just an idea on paper...not reality.

Yes, we know the general expansion rate. But the exact physics is unknown. As I understand it, since the universe was 380,000 years old (and mostly likely long before that), the expansion rate has been accelerating. Which means that the dark energy has been the dominate force since then.

See the dimensions of the universe in my blog.