Classical and Quantum Surgery of Geometries in an Open Inflationary Universe

We study classically and quantum mechanically the Euclidean geometries compatible with an open inflationary universe of a Lorentzian geometry. The Lorentzian geometry of the open universe with an ordinary matter state matches either an open or a closed Euclidean geometry at the cosmological singularity. With an exotic matter state it matches only the open Euclidean geometry and describes a genuine instanton regular at the boundary of a finite radius. The wave functions are found that describe the quantum creation of the open inflationary universe. PACS number(s): 98.80.H, 98.80.Cq, 04.60Ds, 04.60Kz Typeset using REVTEX ∗Electronic address: [email protected] 1 Current data from large-scale structure and the cosmic microwave background suggest open inflationary universe models as a viable theory [1]. Recently Hawking and Turok (HT) proposed such an open inflationary model by using a singular gravitational instanton [2]. This raised, however, a hot debate whether the HT instantons can lead to the most probable wave functions in quantum cosmology [3–5]. Vilenkin argued that the HT-type singular instantons of a massless scalar model may lead to a physically unacceptable consequence [6]. The key issue of the debate is whether one may find the instantons of a Euclidean geometry that are well behaved and match the open inflationary universe of a Lorentzian geometry. Hence, the condition necessary for the surgery of Euclidean and Lorentzian geometries is essential in understanding properly the open inflationary models. In this Letter we study the classical and quantum compatibility condition in gluing Lorentzian and Euclidean geometries. Our model is an open FRW universe of the Lorentzian geometry with a minimal massless scalar field and a cosmological constant. The genuine quantum creation of the Universe requires the backward evolution either to the cosmological singularity or to a gravitational potential barrier at a finite size, in which it fits with a Euclidean universe of the same size. The HT instanton solutions [2,4] and the asymptotically flat solution by Vilenkin [6] belong to the former case. The latter case will also be treated in this paper. Let us begin with an open FRW universe described by the action I = mP 16π ∫