An Open Universe from Inflation

We present a natural scenario for obtaining an open universe ($\Omega _0<1$) through inflation. In this scenario, there are two epochs of inflationary expansion---an epoch of `old inflation,' during which the inflaton field is stuck in a false vacuum, followed by an epoch of `new inflation,' during which the inflaton field slowly rolls toward its true minimum. During the first epoch, inflation solves the smoothness and horizon problems. Then an open universe (with negative spatial curvature) is created by the nucleation of a single bubble. In effect $\Omega$ is instantaneously `reset' to zero. During the subsequent `new' inflation $\Omega$ rises toward unity. The value of $\Omega$ today is calculable in terms of the parameters of the potential, and we show that obtaining values significantly different from zero or unity (though within the range $0<\Omega <1$) does not require significant fine tuning. We compute the spectrum of density perturbations by evolving the Bunch-Davies vacuum modes across the bubble wall into its interior.