There are quite a few challenges for cosmologists trying to discover evidence of what it was like early in the life of our universe, but while the inflationary universe scenario was an almost universally accepted model, there are still problems, especially for observational astrophysicists who don’t see any evidence of a flat universe, as inflation would predict.
What was previously accepted has now created some doubt, especially among the more abstract theorists. Inflation is based around a simple idea. The universe began in a tiny patch of space, which in turn was dominated by the potential energy of some scalar field, a sort of super-dense dark energy. This caused the patch to expand at a terrifically accelerated rate, which smoothed out the density and diluted any strange relics. In time, the scalar field decayed into ordinary matter and radiation that started to reheat the universe into a more conventional status.
Roger Penrose has argued that a universe like ours had to start out in a very particular state and that inflation by itself was not the final answer. It implies that a theory that states the reasons why inflation started is needed. All models that include inflation predict that it will never completely end. The changeable nature of quantum fluctuations even imply that the inflation won’t smooth everything out perfectly. This means that in some places, inflation will curtail while in others, it will continue on forever. This could lead to the beginning of a multiverse, a chaotic jumble consisting of numerous pocket universes separated by regions of inflating spacetime.
While eternal inflation offers solutions, it presents problems. The most important one involves the unitary or Liouville problem. In classical mechanics, the Liouville Theorem states that if you take a number of event states and evolve them forward in time, you’ll end up with the exact number of states you started out with. No new states are created and none are destroyed. If there are a number of states that qualify as the initial conditions necessary for inflation, eternal inflation would evolve them forward into time, and we’d get a collection of universes that would grow in time. The problem arises that as the collection grows, there will be an increasing identical number of states, which didn’t begin as a single, tiny, inflating patch. So while it might be true that you can generate an infinite number of universes, at the same time, the fraction of such states that actually in a single inflating patch quickly goes to zero.
The second problem is known as the measure problem, which tries to calculate the probabilities within the infinite ensemble of universes that eternal inflation creates. There’s a basic problem, we don’t know what kind of “things” we should be counting to calculate the fraction necessary to get the probability, even when using regularization. In the context of eternal inflation, it’s hard to predict anything at all.
The final problem is called the holography/complementarity problem, which arose when physicists thought about black hole entropy and proposed horizon complementarity. This idea stated that one observer couldn’t talk about things happening outside their own horizon. In cosmology, it means that things need to be studied locally, one pocket universe at a time, not all of them.