**5.5. A worked example: polynomial chaotic inflation**

The simplest inflation model
[6]
arises when one chooses a
polynomial potential, such as that for a massive but otherwise
non-interacting field,
*V*() =
*m*^{2}^{2} /
2 where *m* is the mass of the
scalar field. With this potential, the slow-roll equations are

and the slow-roll parameters are

So inflation can proceed provided
|| > *m*_{Pl} /
(4), i.e. as
long as we are not to close to the minimum.

The slow-roll equations are readily solved to give

(50)
(51)

(where
=
_{i} and
*a* = *a*_{i} at *t* = 0) and the
total amount of inflation is

This last equation can be obtained from the solution for *a*, but in
fact is more easily obtained directly by integrating Eq. (47),
for which one needn't bother to solve the equations of motion.

In order for classical physics to be valid we require
*V* << *m*_{Pl}^{4},
but it is still easy to get enough inflation provided *m* is small
enough. As we shall later see, *m* is in fact required to be small from
observational limits on the size of density perturbations produced, and
we can easily get far more than the minimum amount of inflation required
to solve the various cosmological problems we originally set out to
solve.