3.1. The flatness problem
Taking advantage of the definition of the density parameter, and ignoring a possible cosmological constant contribution, the Friedmann equation can be written in the form
During standard big bang evolution,
a2H2 is decreasing, and so
moves away from one, for example
where the solutions apply provided
is close to one. So
= 1
is an unstable critical point. Since we know that today
is
certainly within an order of magnitude of one, it must have been much closer
in the past. Inserting the appropriate behaviours for the matter and
radiation eras (or if you like just assuming radiation domination all
the way to the present) gives
That is, hardly any choices of the initial density lead to a Universe like
our own. Typically, the Universe will either swiftly recollapse, or will
rapidly expand and cool below 3K within its first second of existence.