The Horizon Problem
When we look at the microwave background radiation coming
from opposite parts of the sky, these regions
are too separated to have been able to
have ever communicated with each other even with signals
traveling at light velocity. How do they
know to have the same temperature?
The Flatness Problem
The present Universe has very low geometrical curvature in its
space-time (it is nearly flat). This is a very unlikely result, unless
the initial curvature is confined to an incredibly narrow range of
possibilities. While this is not impossible, it does not seem very natural.
How to deal with these problems? The Inflationary Theory!
The preceding problems with the big bang can be alleviated all at once (at least in principle), by a new kind of cosmology called the inflationary universe.
In this theory, the Universe, expands for a fleeting instant at its beginning at a much higher rate than that expected for the big bang. This period is called the inflationary epoch and happened 10-35 seconds after the creation of the Universe. It filled the Universe with a kind of energy called the vacuum energy and gravitation became repulsive for a period of about 10-32 seconds. During this period the Universe expanded at an astonishing rate, increasing by about a factor of 1050. Then, when this phase transition was complete (akin to water freezing to ice) the universe settled down into the big bang evolution we see today.