**4.2. Solving the Flatness Problem**

How does inflation solve this flatness problem?
How does inflation set up the condition of
= 1?
Consider Eq. 14: (1 - )*H*^{2} *R*^{2} = *constant*.
During inflation *H* =
(_{inf} /
3)^{1/2} = *constant* (Eq. 22) and
the scale factor *R* increases by many orders of magnitude,
10^{30}.
One can then see from Eq. 14 that the large increase in the
scale factor *R* during inflation, with *H* constant, drives
1.
This is what is meant when we say that inflation
makes the Universe spatially flat. In a vacuum-dominated expanding universe,
= 1 is a stable
fixed point.
During inflation *H* is constant and *R* increases exponentially.
Thus, no matter how far
is from 1 before
inflation, the exponential increase of *R* during inflation
quickly drives it to 1 and this is equivalent to flattening the Universe.
Once driven to
= 1 by inflation,
the Universe will naturally evolve away from
= 1 in the absence
of inflation as we showed in the previous section.