**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,
*a*^{2}*H*^{2} 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.