Cosmological scaling solutions of non-minimally coupled scalar fields

On one hand recent cosmological observations, and particuarly the Hubble diagram for Type Ia supernovae [1] have led to the idea that the universe may be dominated by a component with negative pressure [2] and thus that today the universe is accelerating. Such a component can also, if one sticks to the prediction of inflation that Ω = 1, account for the “missing energy”. Yet, many candidates have been proposed like a cosmological constant, a “dynamical” cosmological constant [3], cosmic strings [4] or a spatially homogeneous scalar field rolling down a potential [5]. On the other hand, potentials decreasing to zero for infinite value of the field have been shown to appear in particle physics models (see e.g. [6,7]). For instance, exponential potentials arise in high order gravity [8], in Kaluza-Klein theories which are compactified to produce the four dimensional observed universe [9] or can arise due to nonperturbative effects such as gaugino condensation [10]. Inverse power law potentials can be obtained in models where supersymmetry is broken through fermion condensates [7]. This gives one more theoretical motivation to study the cosmological implications of a field with such potentials. The cosmological solutions with such a field have first been studied by Ratra and Peebles [11] (see also [12,13]) who showed the existence and stability of scaling solutions in respectively a field, radiation or matter dominated dominated universe for a field evolving in an exponential and inverse power law potential. A complete study in the framework of barotropic cosmologies in the case of the exponential potential [14,15] show that the solutions were stable to shear perturbations and to curvature perturbations when P/ρ < −1/3, but that for realistic matter (such as dust) these solutions were unstable, essentialy to curvature perturbations. Liddle and Scherrer [16] made a complete classification of the field potentials and show that power law potentials also lead to scaling solutions (i.e. to solutions such that the field energy density ρφ, behaves as the scale factor at a given power) and the coupling of the field to ordinary matter has been considered in [17]. Such solutions are indeed of interest in cosmology since they provide a candidate for a component with negative pressure. Cosmological models with a scalar field have started to be investigated [18,19] for different kind of potentials as cosine potential [3], exponential potential [20,21] and inverse power law potentials [22]. It has also been shown that the luminosity distance as a function of redshift [23,24] or the behaviour of density perturbations in the dust era as a function of redshift [24] can be used to reconstruct the field potential. However, all these studies have been done under the hypothesis that the field is minimally coupled to the metric. It is known that terms with such a non-minimal coupling R̄f(φ) between the scalar curvature R̄ and the field φ can appear when quantizing fields in curved spacetime [25,26], in multi-dimensional theories [27] like superstring and induced gravity theories [28]. Since these theories predict both the existence of scalar fields with potential or powerlaw potential and non-minimal coupling, it is of interest to study the influence of this coupling, and for instance the robustness of the existence and stability of scaling solutions. The influence of such a coupling during an inflationary period and the existence of inflationnary attractors have yet been examined (see e.g. [29]). In this article, we study the stability of scaling solutions of a non-minimally coupled scalar field. We first present (§II) the main notations and equations. After having, in §III, briefly recalled the standard approach for determining the potentials that can give rise to such behaviour for a minimally coupled scalar field, we investigate the inverse