Scale factor duality and hidden supersymmetry in scalar-tensor cosmology.

It is shown that spatially flat, isotropic cosmologies derived from the Brans– Dicke gravity action exhibit a scale factor duality invariance. This classical duality is then associated with a hidden N = 2 supersymmetry at the quantum level and the supersymmetric quantum constraints are solved exactly. These symmetries also apply to a dimensionally reduced version of vacuum Einstein gravity. PACS NUMBERS: 04.50.+h, 11.30.Pb, 98.80.Hw The possibility that scalar–tensor gravity theories may be relevant in the arena of the very early Universe has been widely investigated in recent years [1, 2, 3, 4, 5]. The defining feature of these theories is the non–minimal coupling of a scalar field to the space–time curvature. These couplings arise naturally in the low energy limit of various unified field theories such as superstring theory [4, 6]. Theories containing higher–order terms in the Ricci scalar may be also expressed in a scalar– tensor form by means of a suitable conformal transformation [7]. Furthermore, the dimensional reduction of higher–dimensional gravity also results in an effective scalar– tensor theory [8]. The study of scalar–tensor theories is therefore well motivated. In this paper we consider the D-dimensional, vacuum theory S = ∫ dx √−ge−Φ [ R − ω (∇Φ) − 2Λ ] , (1) where Φ represents the dilaton field, g is the determinant of the space-time metric G, ω is a space–time constant and Λ is the effective cosmological constant. This theory represents one of the simplest extensions to Einstein gravity and is formally equivalent to the standard Brans-Dicke theory when Λ = 0 [9]. It represents the genus–zero effective action for the bosonic string when the antisymmetric tensor field Bμν vanishes, ω = −1 and Λ = (D − 26)/3α′, where α′ is the inverse string tension [6]. The effective action for the bosonic sector of the closed superstring also has this form, but with Λ proportional to (D − 10). It is known that the reduced string action corresponding to cosmologies that are spatially flat and homogeneous exhibits a symmetry known as scale factor duality [10]. The symmetry group is ZD−1 2 and it is generated by inverting the scale factor and shifting the value of the dilaton. Scale factor duality is a special case of a more general O(d, d) symmetry of the theory and it relates expanding dimensions to contracting ones [11]. It also forms the basis for the pre–big bang cosmological model developed by Gasperini and Veneziano [5]. In general, it is important to search for symmetries in a given theory because they can provide valuable insight into the dynamics of the Universe. They form the basis for selection rules that forbid the existence of certain states and processes. They also allow one to generate new, inequivalent solutions to the field equations from a given solution. Moreover, the conserved quantity associated with a given symmetry represents an integrability condition on the field equations. In some cases, this may allow more general solutions to be found. In this paper we find the analogue of scale factor duality in theory (1) when ω 6= −1. We show that a duality invariance exists for spatially flat, isotropic cosmologies of arbitrary dimension D ≥ 2 when ω 6= −D/(D−1). We then consider the quantization of this family of Universes by mapping the system onto the constrained oscillator– ghost–oscillator model. This procedure uncovers a hidden supersymmetry in the theory that exists at the quantum level. The origin of this supersymmetry may