**5.7. Solving the Problems of the Standard Cosmology**

Let's stick with the simple model of the last section. It is rather easy to see that the Einstein equations are solved by

(213) |

as we might expect in inflation. The period of time during
which (208) is valid ends at
*t* ~ *t*, at
which

(214) |

Now, taking a typical value for *m*, for which
<
10^{-4}, we obtain

(215) |

This has a remarkable consequence. A proper distance
*L*_{P} at *t* = 0
will inflate to a size 10^{108}cm after a time
*t* ~ 5 ×
10^{-36} seconds. It is important at this point to know that
the size of the observable universe today is
*H*_{0}^{-1} ~ *and has not had
enough* 10^{28} cm! Therefore, only a small fraction of the
original Planck
length comprises today's entire visible universe. Thus, homogeneity over a
patch less than or of order the Planck length at the onset of
inflation is all that is required to solve the horizon problem. Of
course, if we wait sufficiently long we will start to see those
inhomogeneities (originally sub-Planckian) that were inflated
away. However, if inflation lasts long enough (typically about sixty
e-folds or so) then this would not be apparent today. Similarly,
any unwanted relics are diluted by the tremendous expansion; so long
as the GUT phase transition happens before inflation, monopoles will
have an extremely low density.

Inflation also solves the flatness problem. There are a couple of ways
to see this. The first is to note that, assuming inflation begins, the
curvature term in the Friedmann equation very quickly becomes
irrelevant, since it scales as *a*(*t*)^{-2}. Of
course, after
inflation, when the universe is full of radiation (and later dust) the
curvature term redshifts more slowly than both these components and
will eventually become more important than both of them. However, if
inflation lasts sufficiently long then even today the total energy
density will be so close to unity that we will notice no curvature.

A second way to see this is that, following a similar analysis to that
leading to (173) leads to the conclusion that
= 1 is an
*attractor* rather than a repeller, when
the universe is dominated by energy with *w* < - 1/3.
Therefore, is
forced to be very close to one by inflation;
if the inflationary period lasts sufficiently long, the density
parameter will not have had time since to stray
appreciably away from unity.