4.1. The flatness problem
Inflation solves the flatness problem more or less by definition (so
that at least any classical, as opposed to quantum, solution of
the problem will fall under the umbrella of the inflationary
definition). From the middle condition, inflation is precisely the
condition that is forced
towards one rather than away from it.
As we shall see, this typically happens very rapidly. A short period of
such behaviour won't do us any good, as the subsequent non-inflationary
behaviour (in particular the standard big bang evolution from
nucleosynthesis onwards) will take us away from flatness again, but all
will be well provided we have enough inflation that
is moved
extremely close to one during the inflationary epoch. If it is close
enough, then it will stay very close to one right to the present,
despite being repelled from one for all the post-inflationary period.
Obtaining sufficient inflation to perform this task is actually fairly
easy. A schematic illustration of this behaviour is shown in
Figure 1.
In the above discussion, I have ignored a possible cosmological constant contribution, but if present it modifies the Friedmann equation to
and so it is
+
which is forced to one. In
general, it is spatial flatness (k
0) that we are driven
towards, not a critical matter density.