Let's set aside the very early stages for the moment. The part covering almost all the history, with a rapid expansion slowing a little bit due to gravity then speeding up a bit also due to gravity, is not really so strange.
In General Relativity (an elegant theory confirmed to great accuracy by many experiments) the effect of gravity between objects is to pull the universe together, slowing expansion. In early times, when the density of objects was high, that was by far the biggest effect, so the initial fast expansion slowed a bit. Still, there was enough expansion so that the density of objects became pretty low. Then the effect of gravity from a background fixed energy density (a cosmological constant) becomes relatively important. In GR, that sort of fixed background "dark energy" gives an accelerating expansion, such as we now have.
The pattern of these changes in expansion rates can be tracked using the red shifts of so-called standard candle stars at various distances from us. They allow us to see into various stages of the past. So that part of the story is fairly definite, although the explanation of the origin of the cosmological constant is very much an open question. It may not even turn out to be best pictured as a dark energy density but perhaps as some sort of interaction with parallel spaces, etc. The reason that such questions are hard to sort out is that the different stories give the same major effects, the ones we just described.
Now let's step back toward the early stages. Something made the universe almost geometrically flat and almost uniform, with tiny density ripples, at the start of the story we just described above. It turns out that a temporary background energy density, which can easily arise in basic theories of fundamental fields, would cause a rapidly accelerating expansion giving all those effects, so long as the acceleration ended smoothly enough. Such "cosmic inflation" is now taken to be the standard picture. It even predicts many of the details of the statistics of the pattern of ripples in the background radiation. As Paul Steinhardt has recently described in Scientific American, a fuller theoretical exploration of inflation indicates that this standard story does not fit easily with the observed large-scale uniformity of our universe. There are alternate accounts of the origin of the nearly uniform, nearly flat state found at the beginning of our main story. The alternate accounts make slightly different predictions than inflation for the ripple statistics. We may learn rather soon which of the theories can be ruled out, thanks to accurate satellite measurements of the background.
These accounts are all grounded in GR, in some cases embedded in versions of string theory. The behavior is obtained from solving the equations. The predictions are checked with detailed observations. Issues like what the expansion rate would be at different times due to the balance of gravity and other effects are already fully included
in all these models. So far as I know, there's no non-mathematical royal road to cosmology.
(published on 12/15/10)