False Vacuum Inflation with a Quartic Potential

We consider a variant of Hybrid Inflation, where inflation is driven by two interacting scalar fields, one of which has a `Mexican hat' potential and the other a quartic potential. Given the appropriate initial conditions one of the fields can be trapped in a false vacuum state, supported by couplings to the other field. The energy of this vacuum can be used to drive inflation, which ends when the vacuum decays to one of its true minima. Depending on parameters, it is possible for inflation to proceed via two separate epochs, with the potential temporarily steepening sufficiently to suspend inflation. We use numerical simulations to analyse the possibilities, and emphasise the shortcomings of the slow-roll approximation for analysing this scenario. We also calculate the density perturbations produced, which can have a spectral index greater than one.