An Open Universe from Inflation

TL;DR

The paper presents a natural two-stage inflationary model—'old inflation' in a false vacuum followed by 'new inflation'—in which a single bubble nucleation inside de Sitter space creates an open FRW interior with negative curvature. After nucleation, a short epoch of slow-roll inflation drives $\Omega$ toward unity, yielding a present-day value $\Omega_0$ that is determined by the potential and reheating scale, allowing a wide range $0<\Omega_0<1$ without fine-tuning. The authors compute the spectrum of density perturbations by propagating Bunch-Davies vacuum fluctuations across the bubble wall into the interior, obtaining explicit expressions for the power spectra $P_{\Phi}(\zeta)$ and $P_{\chi}(\zeta)$ that generalize flat-space results to open universes. They show that an open or near-flat universe can be naturally produced while preserving inflation’s solutions to the horizon and smoothness problems, with distinctive implications for large-scale perturbations and CMB anisotropies. The work also clarifies the role of initial conditions and highlights observational signatures tied to negative spatial curvature, while noting open questions about gravitational waves and the full treatment of wall-thickness effects.

Abstract

We present a natural scenario for obtaining an open universe ($Ω_0<1$) 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 $Ω$ is instantaneously `reset' to zero. During the subsequent `new' inflation $Ω$ rises toward unity. The value of $Ω$ 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 $0<Ω<1$) 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.