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.
Figure 1. A possible evolution of . There may or may not be evolution before inflation, shown by the dotted line. During inflation is forced dramatically towards one, and remains there right up to the present. Only in the extremely distant future will it begin to evolve away from one again.
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.
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.