In the inflationary theory this problem goes away completely, because in contrast to the conventional theory it postulates a period of accelerated expansion while this repulsive gravity is taking place. That means that if we follow our universe backwards in time towards the beginning using inflationary theory, we see that it started from something much smaller than you ever could have imagined in the context of conventional cosmology without inflation. While the region that would evolve to become our universe was incredibly small, there was plenty of time for it to reach a uniform temperature, just like a cup of coffee sitting on the table cools down to room temperature. Once this uniformity is established on this tiny scale by normal thermal-equilibrium processes and I'm talking now about something that's about a billion times smaller than the size of a single proton inflation can take over, and cause this tiny region to expand rapidly, and to become large enough to encompass the entire visible universe. The inflationary theory not only allows the possibility for the universe to be uniform, but also tells us why it's uniform: It's uniform because it came from something that had time to become uniform, and was then stretched by the process of inflation.
The second peculiar feature of our universe that inflation does a wonderful job of explaining, and for which there never was a prior explanation, is the flatness of the universe the fact that the geometry of the universe is so close to Euclidean. In the context of relativity, Euclidean geometry is not the norm; it's an oddity. With general relativity, curved space is the generic case. In the case of the universe as a whole, once we assume that the universe is homogeneous and isotropic, then this issue of flatness becomes directly related to the relationship between the mass density and the expansion rate of the universe. A large mass density would cause space to curve into a closed universe in the shape of a ball; if the mass density dominated, the universe would be a closed space with a finite volume and no edge. If a spaceship traveled in what it thought was a straight line for a long enough distance, it would end up back where it started from. In the alternative case, if the expansion dominated, the universe would be geometrically open. Geometrically open spaces have the opposite geometric properties from closed spaces. They're infinite. In a closed space two lines which are parallel will start to converge; in an open space two lines which are parallel will start to diverge. In either case what you see is very different from Euclidean geometry. However, if the mass density is right at the borderline of these two cases, then the geometry is Euclidean, just like we all learned about in high school.
In terms of the evolution of the universe, the fact that the universe is at least approximately flat today requires that the early universe was extraordinarily flat. The universe tends to evolve away from flatness, so even given what we knew ten or twenty years ago we know much better now that the universe is extraordinarily close to flat we could have extrapolated backwards and discovered that, for example, at one second after the Big Bang the mass density of the universe must have been equal, to an accuracy of 15 decimal places, to the critical density where it counterbalanced the expansion rate to produce a flat universe. The conventional Big Bang theory gave us no reason to believe that there was any mechanism to require that, but it has to have been that way to explain why the universe looks the way it does today. The conventional Big Bang theory without inflation really only worked if you fed into it initial conditions which were highly finely tuned to make it just right to produce a universe like the one we see. Inflationary theory gets around this flatness problem because inflation changes the way the geometry of the universe evolves with time. Even though the universe always evolves away from flatness at all other periods in the history of the universe, during the inflationary period the universe is actually driven towards flatness incredibly quickly. If you had approximately 10-34 seconds or so of inflation at the beginning of the universe, that's all you need to be able to start out a factor of 105 or 1010 away from being flat. Inflation would then have driven the universe to be flat closely enough to explain what we see today.
There are two primary predictions that come out of inflationary models that appear to be testable today. They have to do (1) with the mass density of the universe, and (2) with the properties of the density non-uniformities. I'd like to say a few words about each of them, one at a time. Let me begin with the question of flatness.
The mechanism that inflation provides that drives the universe towards flatness will in almost all cases overshoot, not giving us a universe that is just nearly flat today, but a universe that's almost exactly flat today. This can be avoided, and people have at times tried to design versions of inflation that avoided it, but these versions of inflation never looked very plausible. You have to arrange for inflation to end at just the right point, where it's almost made the universe flat but not quite. It requires a lot of delicate fine-tuning, but in the days when it looked like the universe was open some people tried to design such models. But they always looked very contrived, and never really caught on.
The generic inflationary model drives the universe to be completely flat, which means that one of the predictions is that today the mass density of the universe should be at the critical value which makes the universe geometrically flat. Until three or four years ago no astronomers believed that. They told us that if you looked at just the visible matter, you would see only about one percent of what you needed to make the universe flat. But they also said that they could offer more than that there's also dark matter. Dark matter is matter that's inferred to exist because of the gravitational effect that it has on visible matter. It's seen, for example, in the rotation curves of galaxies. When astronomers first measured how fast galaxies rotate, they found they were spinning so fast that if the only matter present was what you saw, galaxies would just fly apart.