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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(phi) = m2phi2 / 2 where m is the mass of the scalar field. With this potential, the slow-roll equations are

Equation 48 (48)

and the slow-roll parameters are

Equation 49 (49)

So inflation can proceed provided |phi| > mPl / sqrt(4pi), i.e. as long as we are not to close to the minimum.

The slow-roll equations are readily solved to give

Equation 50 (50)
Equation 51 (51)

(where phi = phii and a = ai at t = 0) and the total amount of inflation is

Equation 52 (52)

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 << mPl4, 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.

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