3.7. A worked example: polynomial chaotic inflation

The simplest inflation model [9] arises when one chooses a polynomial potential, such as that for a massive but otherwise non-interacting field, V() = m²² / 2 where m is the mass of the scalar field. With this potential, the slow-roll equations are

(33)

and the slow-roll parameters are

(34)

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

The slow-roll equations are readily solved to give

(35)

(36)

(where = _i and a = a_i at t = 0) and the total amount of inflation is

(37)

This last equation can be obtained from the solution for a, but in fact is more easily obtained directly by integrating Eq. (32), for which one needn't have bothered to solve the equations of motion.

In the older inflationary literature, it was typically assumed that in order for classical physics to be valid, we would require V << m_Pl⁴ but no other restriction would be necessary. In particular, considering scalar field theory in flat space-time, there is no particular meaning to the actual value of the scalar field (which for instance could be shifted by a constant). With those presumptions, we see that while sufficient inflation requires || > m_Pl, one can readily get enough inflation provided m is small enough, and indeed we will see later that m is in fact required to be small from observational limits on the size of density perturbations produced, allowing far more than the minimum amount of inflation required to solve the various cosmological problems we originally set out to solve.