CERN-TH/96-49 hep-th/9602096 BIRTH OF THE UNIVERSE AS QUANTUM SCATTERING IN STRING COSMOLOGY

In a Wheeler-De Witt approach to quantum string cosmology, the present state of the Universe arises from the scattering and reflection of the wave function representing the initial string vacuum in superspace. This scenario is described and compared with the more conventional quantum cosmology picture, in which the birth of the Universe is represented as a process of tunnelling “from nothing” in superspace. Essay written for the 1996 Awards of the Gravity Research Foundation (Wellesley Hills, Ma, 02181-0004) CERN-TH/96-49 February 1996 Permanent address: Dipartimento di Fisica Teorica, Via P. Giuria 1, 10125 Turin, Italy. In the standard cosmological model, the birth of our Universe is assumed to coincide with the initial big-bang singularity, characterizing the classical solutions of the Einstein cosmological equations. Near the singularity, however, the Universe approaches the Planck curvature scale and the quantum gravity regime, where a classical description of the spacetime manifold is no longer appropriate. By adopting, in that regime, a quantum cosmology approach, one can describe the birth of the Universe as a “tunnelling from nothing” [1]-[3], where the process of tunnelling refers to the Wheeler-De-Witt (WDW) wave function [4] in superspace. In string cosmology models [5], the Universe starts evolving from the string perturbative vacuum, a state with flat metric and vanishing coupling constant. The curvature and the coupling grow during an initial “pre-big-bang” phase, and this growth, according to the low-energy effective action, leads classically to a singular state which marks the beginning of the standard, post-big-bang cosmological era. By applying a quantum cosmology approach, the transition through the singular big-bang regime can be described as a scattering of the initial pre-big-bang state into a final post-big-bang configuration, in particular as a reflection of the WDW wave function in superspace. The purpose of this paper is to illustrate this effect and to stress analogies and differences with the more conventional tunnelling scenario for the birth of the Universe. For an easier comparison of the two pictures we shall work with the simplest example of non-trivial WDW equation, in which the effective potential is the one induced by a positive cosmological constant Λ, due to an over-critical number of dimensions. We start with the tree-level, low-energy string effective action [6] S = − 1 2λs ∫ dx √ −ge (R + ∂μφ∂ φ+ Λ) , (1) where φ is the dilaton field, λs is the fundamental string-length parameter governing the derivative expansion of the effective action, and the extra dimensions have been taken to be completely inert. For a homogeneous, isotropic and spatially flat metric background, with scale factor a and spatial sections of finite volume, we define: φ = φ− ln ∫ (dx/λs)− √ 3β, β = √ 3 ln a. (2) In the cosmic-time gauge, g00 = 1, the action (1) becomes: S = − λs 2 ∫ dte ( φ̇ 2 − β̇ + Λ ) . (3)