Boundary conditions in quantum string cosmology

We discuss in detail how to consistently impose boundary conditions in quantum string cosmology. Since a classical time parameter is absent in quantum gravity, such conditions must be imposed with respect to intrinsic variables. Constructing wave packets for minisuperspace models from different tree-level string effective actions, we explain in particular the meaning of a transition between “pre-big-bang” and “post-big-bang” branches. This leads to a scenario different from previous considerations. submitted to Physics Letters B ∗E-mail: [email protected] †E-mail: [email protected] 1 String theory seems to be one of the best current candidates for a theory which unifies gravity with other interactions. Since it applies to energies of the order of the Planck scale, it attracts the interest of cosmologists who are interested in initial conditions for the universe very close to a classical singularity. Much interest has been focused recently on low-energy effective actions from string theory [1]. Such actions contain additional fields in the gravitational sector, in particular dilaton and axion fields. One of the advantages of such an effective theory is the possibility of having a superinflationary phase a(t) ∼ (−t)p (t < 0, p < 0), which is driven by the kinetic energy of the dilaton, and which is free from the fine-tuning problem usually present in potential energydriven de Sitter or power-law inflation. One of the central features of string theory is its symmetry with respect to duality transformations [2]. For simple isotropic cosmologies this leads to the scale factor duality (a → 1/a) [3] which, when combined with time reversal symmetry, results in new, duality-related solutions. Usually, one considers one of these solutions as describing a superinflationary accelerated expansion and the other one as describing a decelerated (presumably radiation dominated) expansion. However, the superinflationary phase emerges only for negative times (t < 0) and its decelerated duality-related branch is separated by a singularity in curvature and string coupling. A desirable scenario would be to have a superinflationary phase for negative times (the “pre-big-bang” phase) followed by a standard radiation dominated expansion (the “post-big-bang” phase). However, in view of the appearance of the singularity between the two phases, this does not seem to be easily achievable. One thus looks for possible mechanisms to overcome this “graceful exit problem” in string cosmology [4]. It has been proven as a new type of “no-go” theorem [5] that there is no way to connect classically the duality-related solutions and to overcome the “graceful exit problem” in the simplest models of string cosmology. With respect to this result, it seems that the classical scenario breaks down and that one needs to take quantum effects into account to avoid the singularity. This can be achieved, for instance, by adopting higher-order α′ (inverse string tension) corrections to the tree-level effective action [1,6]. Such an approach, though 2 preliminary, has been presented recently in [7]. Another possibility is to apply a one-loop superstring effective action, for which there exists a large class of nonsingular solutions for a very broad range of the parameters given in [8]. Staying on the tree-level sector of the string effective action, the formalism of canonical quantum gravity has been applied to describe a quantum transition form the “pre-big-bang” phase to the “post-big-bang” phase through the singularity [9,10]. More precisely, in the minisuperspace comprising scale factor (a) and dilaton (φ), a solution to the Wheeler-DeWitt equation was found after imposing boundary conditions in the strong coupling regime φ→ ∞. This solution was interpreted as describing a reflection in minisuperspace through the singularity. Such an interpretation is, however, tight to the presence of an external time parameter. Being redundant already in the classical theory due to time-reparametrisation invariance, an external time parameter is completely absent from quantum gravity (see, for example, the careful discussion in [11,12]). A classical time parameter can only emerge as an approximate notion through some Born-Oppenheimer type of expansion scheme [13]. How, then, can the above “transition” be consistently dealt with in quantum cosmology? The choice of boundary conditions as well as the interpretation of the quantum cosmological wave function should refer only to intrinsic variables, i.e. variables which directly occur in the Wheeler-DeWitt equation. In this respect the hyperbolic nature of this equation for such models is particularly important [11]. The purpose of our paper is the presentation of a consistent quantum cosmological scenario along these lines. Instead of referring to an external time, we shall construct wave packets that represent classical trajectories in quantum cosmology. This has been successfully applied before in quantum general relativity [14]. Furthermore, we shall suggest to impose boundary conditions in the region of small scale factor. In the following we shall first stick to the simple model where only a positive cosmological constant is present [9,10]. The main conceptual issues can be discussed clearly in this context. We shall then proceed to discuss an example which exhibits turning points in configuration 3 space. Before starting with the details of our analysis, we would like to emphasise that it is not, in general, justified to quantise an effective action (which itself arises from a fundamental quantum theory). For example, one would certainly not invoke a quantisation of the “EulerHeisenberg” effective action of QED. However, in so far as new fundamental fields arise from the fundamental theory (such as dilatons and axions), a quantisation of the effective action could capture some relevant features. It is with this reservation that we present the following investigation. We start with the Wheeler-De Witt (WDW) equation from a tree-level low-energy string effective action for zero spatial curvature, which contains a dilaton potential similarly to [9,10]. (Quantum cosmology for string models was first studied in [15].) It reads ĤΨ ≡ [ −∂ φ̄ + ∂ β − λsV (β, φ̄)e−2φ̄ ]